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Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – Number, Number Sense and Operations Standard Students demonstrate number sense, including 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 A. Use scientific notation to express large numbers and numbers less than one. 1. Demonstrate an understanding of place value using powers of 10 and write large numbers in scientific notation. (Grade 7) Number and Number Systems 1. Use scientific notation to express large numbers and small numbers between 0 and 1. B. Identify subsets of the real number system. 1. scientific notation – a form of writing numbers as the product of a power of 10 and a decimal number greater than or equal to 1 and less than 10 3. Describe differences between rational and irrational numbers, such as use technology to show that some numbers (rational) can be expressed as terminating or repeating decimals and others (irrational) as non-terminating and non-repeating decimals. (Grade 7) Number and Number Systems 2. Recognize that natural numbers, whole numbers, integers, rational numbers and irrational numbers are subsets of the real number system. C. Apply properties of operations and the real number system and justify when they hold a set of numbers. Meaning of Operations F. Explain the effects of operations on the magnitude of quantities. 10. Recognize that a quotient may be larger than the dividend when the divisor is a fraction, such as 6 ÷ ½ = 12. (Grade 6) 4. Explain and use the inverse and identity properties and use inverse relationships (addition/subtraction, multiplication/division, squaring/square roots) in problemsolving situations. 2. integers – set of whole numbers and their opposites rational numbers – number that can be expressed as a ratio of two integers irrational numbers – a number that cannot be represented by a ratio of two integers 5. Explain the meaning and effect of adding, subtracting, multiplying and dividing integers, such as how adding two integers can result in a lesser value. (Grade 7) Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 125 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – Number, Number Sense and Operations Standard (continued) Students demonstrate number sense, including 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 G. Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions. Computation and Estimation 5. Determine when an estimate is sufficient and when an exact answer is needed in problem situations, and evaluate estimates in relation to actual answers, such as very close, less than, greater than. 6. Estimate, compute and solve problems involving rational numbers, including ratio, proportion and percent, and judge the reasonableness of solutions. 5. Recognize and identify perfect squares and their roots. (Grade 5) H. Find the square root of perfect squares, and approximate the square root of non-perfect squares. Computation and Estimation 7. Find the square root of perfect squares, and approximate the square root of nonperfect squares as consecutive integers between which the root lies, such as √130 is between 11 and 12. 2. Explain the meaning of exponents that are negative or 0. (Grade 7) I. Estimate, compute and solve problems involving scientific notation, square roots and numbers with integer exponents. Meaning of Operations 3. Apply order of operations to simplify expressions and perform computations involving integer exponents and radicals. Computation and Estimation 8. Add, subtract, multiply, divide and compare numbers written in scientific notation. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 126 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – 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 A. Solve increasingly complex non-routine measurement problems and check for reasonableness of results. Analyze problem situations involving measurement concepts, select appropriate strategies and use an organized approach to solve narrative and increasingly complex problems.(Grade 7) Use Measurement Techniques and Tools 6. Solve and determine the reasonableness of the results for problems involving rates and derived measurements, such as velocity and density, using formulas, models and graphs. B. Use formulas to find surface area and volume for specified three-dimensional objects accurate to a specified level of precision. Use Measurement Techniques and Tools C. Apply indirect measurement techniques, tools and formulas, as appropriate, to find perimeter, circumference and area of circles, triangles, quadrilaterals and composite shapes and to find volume of prisms, cylinders and pyramids. Use Measurement Techniques and Tools 3. Use appropriate levels of precision when calculating with measurements. 3. absolute error relative error 4. Derive formulas for surface area and volume and justify those using geometric models and common materials. For example, find: a. the surface area of a cylinder as a function of its height and radius; b. that the volume of a pyramid (or cone) is one-third of the volume of a prism (or cylinder) with the same base area and height. 5. Determine surface area for pyramids by analyzing their parts. 9. Demonstrate understanding of the concepts of perimeter, circumference and area by using established formula for triangles, quadrilaterals, and circles to determine the surface area and volume of prisms, pyramids, cylinders, spheres and cones. (Note: Only volume should be calculated for spheres and cones). Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 127 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – Measurement Standard (continued) 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 1. Compare and order the relative size of common U.S. customary units and metric units, such as mile and kilometer, gallon and liter, pound and kilogram. 2. Use proportional relationships and formulas to convert units from one measurement system to another, such as degrees Fahrenheit to degrees Celsius. Use Measurement Techniques and Tools 7. Apply proportional reasoning to solve problems involving indirect measurements or rates. E. Estimate and compute various attributes, including length, angle measure, area, surface area and volume, to a specified level of precision. F. Write and solve realworld, multi-step problems involving money, elapsed time and temperature, and verify reasonableness of solutions. 7. similar figures trigonometry ratios Use Measurement Techniques and Tools 3. Use appropriate levels of precision when calculating with measurement. 8. Find the sum of the interior and exterior angles of regular convex polygons with and without measuring the angles with a protractor. 10. Use conventional formulas to find the surface area and volume of prisms, pyramids and cylinders and the volume of spheres and cones to a specified level of precision. Use Measurement Techniques and Tools 6. Solve and determine the reasonableness of the results for problems involving rates and derived measurements, such as velocity and density, using formulas, models and graphs. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 128 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – 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 B. Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence. Characteristics and Properties C. Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines and parallel lines. Characteristics and Properties 1. Make and test conjectures about characteristics and properties, such as sides, angles, symmetry, of two-dimensional figures and three-dimensional objects. 3. Use proportions in several forms to solve problems involving similar figures (partto-part, part-to-whole, corresponding sides between figures). 2. Recognize the angles formed and the relationship between the angles when two lines intersect and when parallel lines are cut by a transversal. D. Use coordinate geometry Characteristics and Properties to represent and examine the 1. Make and test conjectures about characteristics and properties, such as sides, angles, symmetry, of two-dimensional figures and three-dimensional objects. properties of geometric figures. Spatial Relationships 4. Represent and analyze shapes using coordinate geometry, such as given three vertices and the type of quadrilateral, find the coordinate of the fourth vertex. Visualization and Geometric Models E. Draw and construct 6. Draw nets for a variety of prisms, pyramids, cylinders and cones. representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 6. a two-dimensional representation of a threedimensional figure for net templates of various solids 129 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – 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 F. Represent and model transformations in a coordinate plane and describe the results. Transformations and Symmetry 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 5. Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed, such as lengths of sides remain the same under translations. 1. Make and test conjectures about characteristics and properties, such as sides, angles, symmetry, of two-dimensional figures and three-dimensional objects. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 130 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade - 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 B. Identify and classify functions as linear or nonlinear, and contrast their properties using tables, graphs or equations. Use Patterns, Relations and Functions C. Translate information from one representation (words, table, graph or equation) to another representation of a relation or function. Use Patterns, Relations and Functions D. Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations. Use Algebraic Representation 2. Generalize patterns and sequences by describing how to find the nth term. 3. Identify functions as linear or nonlinear based on information given in a table, graph or equation. 3. linear equation – an equation that when graphed, forms a straight line 1. Relate the various representations of a relationship, must include relate a table to graph, description and symbolic form. 4. Extend the uses of variables to include covariants where y depends on x. 5. Use physical models to add and subtract monomials and polynomials, and to multiply a polynomial by a monomial. 7. Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems. 8. Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 131 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade - 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 Algebraic Representation F. Solve and graph linear equations and inequalities. Use Algebraic Representation 6. Describe the relationship between the graph of a line and its equation, including being able to explain the meaning of slope as a constant rate of change and y-intercept in real-world problems. 6. If the slope and the y-intercept of a line are known, the equation can be written in standard form. 7. Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems. 9. Solve linear equations and inequalities graphically, symbolically and using technology. G. Solve quadratic equations with real roots by graphing, formula and factoring. H. Solve systems of linear equations involving two variables graphically and symbolically. I. Model and solve problem situations involving direct and inverse variation. Use Algebraic Representation 12. graph of a quadratic equation is a symmetric curve (parabola) 12. Solve simple quadratic equations graphically, such as y = x² - 16. Use Algebraic Representation 10. Solve 2 x 2 systems of linear equations graphically and by simple substitution. 11. Interpret the meaning of the solution of a 2 by 2 system of equations, such as point, line no solution. Analyze Change 14. Differentiate and explain types of changes in mathematical relationships, such as linear vs. non-continuous, direct variation vs. inverse variation. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 132 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade - 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 J. Describe and interpret rates of change from graphical and numerical data. Use Algebraic Representation 13. Compute and interpret slope, midpoint and distance given a set of ordered pairs. Analyze Change 15. Describe and compare how changes in an equation affect the related graphs, such as for a linear equation changing the coefficient of x affects the slope and changing the constant affects the intercepts. 16. Using graphing calculators or computers to analyze change, such as interest compounded over time as a nonlinear growth pattern. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 16. simple interest, compound interest 133 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – 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. B. Evaluate different graphical representations of the same data to determine which is the most appropriate representation for an identified purpose. Data Collection 1. Use, create and interpret scatterplots and other types of graphs as appropriate. Data Collection 2. Evaluate different graphical representations of the same data to determine which is the most appropriate representation for an identified purpose, such as line graph for change over time, circle graph for part-to-whole comparison, scatterplot for relationship between two variants. 2. graphs that compare – single bar graphs, double bar graphs graphs that show clusters of data information – line plots, histograms, box-and-whisker plots, stem-and-leaf plots, double stem-and-leaf plots 3. Differentiate between discrete and continuous data and appropriate ways to represent graphs that show change over each. time – single line graphs, multiple line graphs 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. Statistical Methods 5. Explain the mean’s sensitivity to extremes and its use in comparison with the median 5. outlier – an outlier can skew the information of a set of data when and mode. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. finding the mean 134 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – 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 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. Statistical Methods E. Evaluate the validity of claims and predictions that are based on data by examining the appropriateness of the data collection and analysis. Statistical Methods F. Construct convincing arguments based on analysis of data and interpretation of graphs. Statistical Methods 4. Compare two sets of data using measures of center (mean, mode, median) and measures of spread (range, quartile, interquartile range, percentiles). 8. Describe how the relative size of a sample compared to the target population affects the validity of predictions. 6. Make conjectures about possible relationship in a scatterplot and approximate line of best fit. 9. Construct convincing arguments based on analysis of data and interpretations of graphs. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 135 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – 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. J. Compute the probabilities of compound events, independent events and simple dependent events. Statistical Methods 7. Identify different ways of selecting samples, such as survey response, random sample, representative sample and convenience sample. 7. random sample – members of population have an equal chance of being selected for the survey representative sample – has about the same proportions as the population being studied Probability 10. Calculate the number of possible outcomes for a situation, recognizing and accounting for when items may occur more than once or when order is important. convenience sample – that portion of the population that are available for the survey. This method may have biased results. 7. Design an experiment to test a theoretical probability and explain how the results may vary. (Grade 6) 8. Make predictions based on theoretical probabilities, design and conduct an experiment to test the predictions, compare actual results to predicted results and explain differences. (Grade 7) Probability 11. Demonstrate an understanding that the probability of either of two disjoint events occurring can be found by adding the probabilities for each and that the probability of one independent event following another can be found by multiplying the probabilities. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. 136 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – 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. B. Apply mathematical knowledge and skills routinely in other content areas and practical situations. C. Recognize and use connections between equivalent representations and related procedures for a mathematical concept; such as, zero of a function and the x-intercept of the graph of the function, apply proportional thinking when measuring, describing functions and comparing probabilities. D. Apply reasoning processes and skills to construct logical verifications or counter-examples to test conjectures and to justify and defend algorithms and solutions. E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas. Note: Letters/numbers correlate to the Ohio Academic Content Standards book. Specific grade-level indicators have not been included for the mathematical processes standard because content and processes should be interconnected at the indicator level. Therefore, mathematical processes have been embedded within the grade-level indicators for the five content standards. 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 137 Muskingum Valley ESC Standards-Based Mathematics Course of Study 2003 Eighth Grade – Mathematical Processes Standard (continued) 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 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. Specific grade-level indicators have not been included for the mathematical processes standard because content and processes should be interconnected at the indicator level. Therefore, mathematical processes have been embedded within the grade-level indicators for the five content standards. Other Stated Verbs in the Indicators: use derive simplify apply demonstrate evaluate explain make differentiate estimate test create compute draw construct solve relate calculate find generalize order extend Problem-Solving Strategies choose a method of computation draw a diagram guess and check look for a pattern make a list make a model solve a simple problem use logical reasoning work backward eliminate possibilities reasonable answers 138