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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Number, Number Sense and Operations Standard Students demonstrate number sense, include an understanding of number systems and operations and how they relate to one another. Students compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 C. Apply properties of operations and the real number system and justify when they hold for a set of numbers. Number and Number Systems D. Connect physical, verbal and symbolic representations of integers, rational numbers and irrational numbers. Number and Number Systems E. Compare, order and determine equivalent forms of real numbers. F. Explain the effects of operations on the magnitude of quantities. G. Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions. (Gr. 9#1) Identify and justify whether properties (closure, identity, inverse, commutative and associative) hold for a given set and operations; such as, even integers and multiplication. (Gr.10#1) Connect physical, verbal and symbolic representations of irrational numbers; such as, construct √2 as a hypotenuse or on a number line. Meaning of Operations (Gr.10#2) Explain the meaning of the nth root. Number and Number Systems (Gr.9#2) Compare, order and determine equivalent forms for rational and irrational numbers. Meaning of Operations (Gr.9#3) Explain the effects of operations, such as multiplication or division, and of computing powers and roots on the magnitude of quantities. Computation and Estimation (Gr.9#4) Demonstrate fluency in computations using real numbers. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. (9#4) Elements of computational fluency: efficiency accuracy fluency 212 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Number, Number Sense and Operations Standard (continued) Students demonstrate number sense, include an understanding of number systems and operations and how they relate to one another. Students compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 I. Estimate, compute and solve problems involving scientific notation, square roots and numbers with integer exponents. Computation and Estimation (Gr.9#5) Estimate the solutions for problem situations involving square and cube roots. (9#5) Always use ≈ instead of = when numbers are approximate. (Gr.10#4) Approximate the nth root of a given number greater than zero between consecutive integers when n is an integer; such as, the 4th root of 50 is between 2 and 3. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 213 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Measurement Standard Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 D. Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates. Measurement Units (Gr.9#1) Convert rates within the same measurement system; such as, miles per hour to feet per second; kilometers per hour to meters per second. Use Measurement Techniques and Tools (Gr.9#2) Use unit analysis to check computations involving measurement. (Gr.9#3) Use the ratio of lengths in similar two-dimensional figures or threedimensional objects to calculate the ratio of their areas or volumes respectively. (Gr.9#4) Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures. (Gr.9#5) Solve problems involving unit conversion for situations involving distances, areas, volumes and rates within the same measurement system. (9#4) scale drawing – drawing that is similar but is either larger or smaller than the original object – example of a scale drawing is a map. (Gr.10#5) Determine the measures of central and inscribed angles and their associated major and minor arcs. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 214 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Geometry and Spatial Sense Standard Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve problems. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 A. Formally define geometric figures. Characteristics and Properties (Gr.10#1) Formally define and explain key aspects of geometric figures, including: a. interior and exterior angles of polygons; b. segments related to triangles (median, altitude, midsegment); c. points of concurrency related to triangles (centroid, incenter, orthocenter and circumcenter); d. circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle). midsegment = midline Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. (Gr.10#2) Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane. Transformations and Symmetry (Gr.10#6) Identify the reflection and rotation symmetries of two- and threedimensional figures. Visualization and Geometric Models (Gr.10#10) Solve problems involving chords, radii and arcs within the same circle. (10#10) chord – line segment that joins two points on a circle radii – (plural of radius) arc – part of a curve between any two points C. Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines and parallel lines. Characteristics and Properties (Gr.8#2) Recognize the angles formed and the relationship between the angles when two lines intersect and when parallel lines are cut by a transversal. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 215 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Geometry and Spatial Sense Standard (continued) Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve problems. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 E. Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology. Spatial Relationships (Gr.10#4) Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software. (Gr.10#5) Construct congruent or similar figures using tools, such as compass, straightedge and protractor or dynamic geometry software. Transformations and Symmetry (Gr.10#7) Perform reflections and rotations using compass and straightedge constructions and dynamic geometry software. F. Represent and model transformations in a coordinate plane and describe the results. G. Prove or disprove conjectures and solve problems involving twoand three-dimensional objects represented within a coordinate system. (10#4-5) Provide experiences with all tools to construct various figures. ex. compass, straightedge, protractor and dynamic geometry as appropriate. Transformations and Symmetry (Gr.10#8) Derive coordinate rules for translations, reflections and rotations of geometric figures in the coordinate plane. (10#7) Use graph paper. (10#8) Distance between an object and its translated, reflected or rotated image can be shown and calculated if images are placed in a coordinate plane. Visualization and Geometric Models (Gr.9#3) Analyze two-dimensional figures in a coordinate plane; such as, use slope and distance formulas to show that a quadrilateral is a parallelogram. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 216 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Geometry and Spatial Sense Standard (continued) Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects and transformations to analyze mathematical situations and solve problems. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 H. Establish the validity of conjectures about geometric objects, their properties and relationships by counterexample, inductive and deductive reasoning, and critiquing arguments made by others. Characteristics and Properties (Gr.10#3) Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: a. prove the Pythagorean Theorem; b. prove theorems involving triangle similarity and congruence; c. prove theorems involving properties of lines, angles, triangles and quadrilaterals; d. test a conjecture using basic constructions made with a compass and straightedge or technology. Visualization and Geometric Models (Gr.10#10) Solve problems involving chords, radii and arcs within the same circle. I. Use right triangle trigonometry relationships to determine lengths and angle measures. Characteristics and Properties (Gr.9#1) Define the basic trigonometric ratios in right triangles: sine, cosine and tangent. (Gr.9#2) Apply proportions and right triangle trigonometric ratios to solve problems involving missing lengths and angle sizes in similar figures. (9#2) mnemonic device: SOHCAHTOA sine = opposite over hypotenuse cosine = adjacent over hypotenuse tangent = opposite over adjacent Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 217 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series - Patterns, Functions and Algebra Standard Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations, such as tables, graphs and equations. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 A. Generalize and explain patterns and sequences in order to find the next term and the nth term. Use Patterns, Relations and Functions (9#2) tabular = table (Gr.9#2) Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations. Use a function table to help identify the pattern in a sequence. (9#1) domain – set of values of the independent variable of a function B. Identify and classify functions as linear or nonlinear, and contrast their properties using tables, graphs or equations. Use Patterns, Relations and Functions (Gr.9#1) Define function with ordered pairs in which each domain element is assigned exactly one range element. range – possible values for y in a function function – relation when the value of x has a value of y (Gr.9#3) Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations. (Gr.10#1) Define function formally and with f(x) notion. (Gr.10#2) Describe and compare characteristics of the following families of functions: square root, cubic, absolute value and basic trigonometric functions; such as, general shape, possible number of roots, domain and range. C. Translate information from one representation (words, table, graph or equation) to another representation of a relation or function. Use Patterns, Relations and Functions (Gr.9#2) Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 218 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series - Patterns, Functions and Algebra Standard (continued) Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations, such as tables, graphs and equations. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 D. Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations. Use Algebraic Representation (Gr.9#7) Use formulas to solve problems involving exponential growth and decay. (Gr.9#11) Add, subtract, multiply and divide monomials and polynomials (division of polynomials by monomials only. (Gr.9#12) Simplify rational expressions by eliminating common factors and applying properties of integer exponents. (Gr.10#3) Solve equations and formulas for a specified variable; such as, express the base of a triangle in terms of the area and height. (Gr.10#4) Use algebraic representations and functions to describe and generalize geometric properties and relationships. (Gr.10#5) Solve simple linear and nonlinear equations and inequalities having square roots as coefficients and solutions. (Gr.10#6) Solve equations and inequalities having rational expressions as coefficients and solutions. (Gr.10#10) Solve real-world problems that can be modeled using linear, quadratic exponential or square root functions. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 219 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series - Patterns, Functions and Algebra Standard (continued) Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations, such as tables, graphs and equations. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 E. Analyze and compare functions and their graphs using attributes, such as,rates of change, intercepts and zeros. Use Patterns, Relations and Functions F. Solve and graph linear equations and inequalities. Use Algebraic Representation (Gr.9#4) Demonstrate the relationship among zeros of a function, roots of equations and solutions of equations graphically and in words. (Gr.9#5) Describe and compare characteristics of the following families of functions; linear, quadratic and exponential functions; such as, general shape, number of roots, domain, range, rate of change, maximum or minimum. (Gr.9#6) Write and use equivalent forms of equations and inequalities in problem situations; such as, changing a linear equation to the slope-intercept form. (9#6) y = mx + b m is the slope b is the y-intercept (Gr.9#8) Find linear equations that represent lines that pass through a given set of ordered pairs, and find linear equations that represent lines parallel or perpendicular to a given line through a specific point. (Gr.10#10) Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions. G. Solve quadratic equations with real roots by graphing, formula and factoring. Use Algebraic Representation (Gr.10#8) Graph the quadratic relationship that defines circles. (Gr.10#10) Solve real-world problems that can be modeled using linear, quadratic, exponential or square root functions. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 220 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series - Patterns, Functions and Algebra Standard (continued) Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities. Students analyze, model and solve problems using various representations, such as tables, graphs and equations. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 H. Solve systems of linear equations involving two variables graphically and symbolically. Use Algebraic Representation (Gr.9#9) Solve and interpret the meaning of 2 by 2 systems of linear equations graphically, by substitution and by elimination, with and without technology. (Gr.10#7) Solve systems of linear equalities. (10#7) A system of linear equations can have one solution, many solutions or no solution. (Gr.10#11) Solve real-world problems that can be modeled, using systems of linear equations and inequalities. I. Model and solve problem situations involving direct and inverse variation. Analyze Change (Gr.9#13) Model and solve problems involving direct and inverse variation using proportional reasoning. (9#13) direct variation y = kx inverse variation y = k x k is variation constant (Gr.9#14) Describe the relationship between slope and the graph of a direct variation and inverse variation. J. Describe and interpret rates of change from graphical and numerical data. Use Algebraic Representation (Gr.10#9) Recognize and explain that the slopes of parallel lines are equal and the slopes of perpendicular lines are negative reciprocals. Analyze Change (Gr.9#15) Describe how change in the value of a constant in a linear or quadratic equation affects the related graphs. (9#15) slope = change in y change in x (Gr.10#12) Describe the relationship between slope of a line through the origin and the tangent function of the angle crated by the line and the positive x-axis. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 221 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Data Analysis and Probability Standard Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 A. Create, interpret and use graphical displays and statistical measures to describe data; such as, boxand-whisker plots, histograms, scatterplots, measures of center and variability. Data Collection (Gr.9#1) Classify data as univariate (single variable) or bivariate (two variables) and as quantitative (measurement) or qualitative (categorical) data. (Gr.9#2) Create a scatterplot for a set of bivariate data, sketch the line of best fit and interpret the slope of the line of best fit. (Gr.10#2) Represent and analyze bivariate data using appropriate graphical displays (scatterplots, parallel box-and-whisker plots, histograms with more than one set of data, tables, charts, spreadsheets) with and without technology. (Gr.10#3) Display bivariate data where at least one variable is categorical. (Gr.10#4) Identify outliers on a data display; such as, use the interquartile range to identify outliers on a box-and-whisker plot. Statistical Methods (Gr.9#3) Analyze and interpret frequency distributions based on spread, symmetry, skewness, clusters and outliers. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 222 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Data Analysis and Probability Standard (continued) Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 C. Compare the characteristics of the mean, median and mode for a given set of data, and explain which measure of center best represents the data. D. Find, use and interpret measures of center and spread, such as mean and quartiles, and use those measures to compare and draw conclusions about sets of data. E. Evaluate the validity of claims and predictions that are based on data by examining the appropriateness of the data collection and analysis. Data Collection (10#1) mean, median, mode (Gr.10#1) Describe measures of center and the range verbally, graphically and algebraically. Statistical Methods (Gr.10#6) Interpret the relationship between two variables using multiple graphical displays and statistical measures; such as, scatterplots, parallel box-and-whisker plots and measures of center and spread. Statistical Methods (Gr.9#4) Describe and compare various types of studies (survey, observation, experiment), and identify possible misuses of statistical data Statistical Methods F. Construct convincing arguments based on analysis of data and interpretation of graphs. (Gr.9#6) Make inferences about relationships in bivariate data, and recognize the difference between evidence of relationship (correlation) and causation. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 223 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Data Analysis and Probability Standard (continued) Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 G. Describe sampling methods and analyze the effects of method chosen on how well the resulting sample represents the population. H. Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes. I. Design an experiment to test a theoretical probability, and record and explain results. Statistical Methods (Gr.9#5) Describe characteristics and limitations of sampling methods, and analyze the effects of random versus biased sampling; such as, determine and justify whether the sample is likely to be representative of the population. (Gr.10#5) Provide examples and explain how a statistic may or may not be an attribute of the entire population; such as, intentional or unintentional bias may be present. (10#5) Bias (intentional or unintentional) may not be representative of the population if a large number of responders did not tell the truth. ex. amount of time students spend on homework Probability (9#7) Fundamental Counting Principle – one can find the total number of ways that two or more separate things can happen by multiplying the number of ways each thing can occur (Gr.9#7) Use counting techniques and Fundamental Counting Principle to determine the total number of possible outcomes for mathematical situations. Probability (Gr.9#8) Describe, create and analyze a sample space and use it to calculate probability. (9#8) sample space – all the possible outcomes of an experiment ex. roll of a number cube – there are 6 possible outcomes A list or tree diagram can be used to show all possible outcomes of an experiment. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 224 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Data Analysis and Probability Standard (continued) Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data. Ohio Benchmarks Grade-Level Indicators Resources/Vocabulary Grades 8-10 J. Compute the probabilities of compound events, independent events and simple dependent events. K. Make predictions based on theoretical probabilities and experimental results. Probability (Gr.9#9) Identify situations involving independent and dependent events, and explain differences between, and common misconceptions about, probabilities associated with those events. (9#9) dependent event – occurrence of one event will have an effect on the probability that a second event will occur (Gr.10#7) Model problems dealing with uncertainty with area models (geometric probability). independent event – occurrence of one event has no effect on the probability that a second event will occur Probability Illustrate situations with concrete objects to assist students in distinguishing between dependent and independent events. ex. spinner (Gr.9#10) Use theoretical and experimental probability, including simulations or random numbers, to estimate probabilities and to solve problems dealing with uncertainty; such as, compound events, independent events and simple dependent events. (Gr.10#8) Differentiate and explain the relationships between the probability of an event and the odds of an event, and compute one given the other. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 225 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Integrated Series – Mathematical Processes Standard Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas. The benchmarks for mathematical processes articulate what students should demonstrate in problem-solving, representation, communication, reasoning and connections at key points in their mathematics program. Ohio Benchmarks Grade-Level Resources/Vocabulary Grades 8-10 Indicators A. Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information and set limits for acceptable solution. Specific grade-level indicators have not been included for the mathematical B. Apply mathematical knowledge and skills routinely in other content processes standard areas and practical situations. because content and processes should be C. Recognize and use connections between equivalent representations and related procedures for a mathematical concept; such as, zero of a function interconnected at the indicator level. and the x-intercept of the graph of the function, apply proportional Therefore, thinking when measuring, describing functions and comparing mathematical probabilities. processes have been D. Apply reasoning processes and skills to construct logical verifications embedded within the or counter-examples to test conjectures and to justify and defend grade-level indicators algorithms and solutions. for the five content standards. E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas. F. Use precise mathematical language and notations to represent problem situations and mathematical ideas. G. Write clearly and coherently about mathematical thinking and ideas. H. Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. compare: to determine how two things are alike and/or different; the common/critical attributes must be identified Compare is involved in ALL of the following: analyze: to investigate by breaking it down so as to more clearly understand the impact to the situation describe: to analyze into its parts but less detailed than explain determine: to reach a decision after a thorough investigation; to find the cause of and then to solve or set limits to a situation identify: to show or prove the sameness of interpret: a student must first analyze and then make an inference; this is more subjective than an evaluation predict: to state what one believes will happen (based on data) recognize: to examine closely and identify the common/critical attributes Other Stated Verbs in the Indicators: justify define connect construct explain derive order test demonstrate apply estimate generalize approximate add convert subtract use multiply solve divide simplify write find graph model classify create display differentiate 226