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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011 Mathematics SOLs/Common Core State Standards Overview for Grades 6—8 Complied by Sharon Antal, Ericca Dent, Courtney Mann, Rob Schupbach, and Holly Stainback
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011 SOL/Common Core Overview a) They both stress application to real-life situations. Both focus on using calculators and appropriate technology (computers and spreadsheets) b) The SOLs focus on problem solving and is a major part of the math program at every grade level, but the Common Core does not mention this.The SOLs encourage students to apply information to science and other disciplinesthey are studying and the common does not mention this.The Common Core mentions that students should be able to develop their ownarguments and reasoning and critique the reasoning of others and the SOLs do notmention this.The SOLS focus using concrete materials for all age levels where the CommonCore suggest their use only for younger children. c) SOL Focus6th GradeFocuses on transitioning from whole number arithmetic to foundations of algebra7th GradeFocuses on continued emphasis of foundations of algebra8th GradeFocuses on reviewing concepts and skills learned in previous grades and givingnew content that prepares students for algebra and geometry d) Common Core Focus Grades 6 - 8In Grade 6, instructional time should focus on four critical areas: (1) connectingratio and rate to whole number multiplication and division and using concepts ofratio and rate to solve problems; (2) completing understanding of division offractions and extending the notion of number to the system of rational numbers,
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011which includes negative numbers; (3) writing, interpreting, and using expressionsand equations; and (4) developing understanding of statistical thinking.In Grade 7, instructional time should focus on four critical areas: (1) developingunderstanding of and applying proportional relationships; (2) developingunderstanding of operations with rational numbers and working with expressionsand linear equations; (3) solving problems involving scale drawings and informalgeometric constructions, and working with two- and three-dimensional shapes tosolve problems involving area, surface area, and volume; and (4) drawinginferences about populations based on samples.In Grade 8, instructional time should focus on three critical areas: (1) formulatingand reasoning about expressions and equations, including modeling an associationin bivariate data with a linear equation, and solving linear equations and systems oflinear equations; (2) grasping the concept of a function and using functions todescribe quantitative relationships; (3) analyzing two- and three-dimensional spaceand figures using distance, angle, similarity, and congruence, and understandingand applying the Pythagorean Theorem. e) Overall, I think the core is a little too specifiic, even when the full scope and sequence of the SOLs are considered. The core seems to talk a good game about abstract reasoning, but I don’t see any proof that there standards are really guaranteeing that this is happening. It’s not exactly there with the SOLs either, but I think there is more of a conceptual definition for certain standards, rather than just formulas (perhaps by design) and as such it seems like at least due to to what form its creator have chosen to write it, a teacher might be able to fit true problem-solving in more easily. However, the specificity with which the Core is written and its insistence on maintaining certain concepts throughout (rational numbers seem to be throughout the grade band and not just put in with a vengeance in 8th grade) the Core might make students really focus on certain concepts and might make teachers better aware of what they should be stressing. Clarity can be really good.From Education Week, September 2011:
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011Fresh PerspectiveThe common standards represent such a big change that states shouldnt even try to findcommonalities between them and their old state standards, but view them "as somethingcompletely new," said Ken Krehbiel, the associate executive director of communications for theNational Council of Teachers of Mathematics.That way, he said, educators are more likely to find fresh ways to design lessons based on the"mathematical practices" that form the core of the standards, emphasizing skills such as problem-solving and mathematical modeling.NCTM and others are working to build a library of sample tasks that reflect those practices andare making them available to educators as they are assembled, Mr. Krehbiel said. Mathematics Standards of Learning for Virginia Public SchoolsGrade SixThe sixth-grade standards are a transition from the emphasis placed on whole number arithmeticin the elementary grades to foundations of algebra. The standards emphasize rational numbers.Students will use ratios to compare data sets; recognize decimals, fractions, and percents asratios; solve single-step and multistep problems, using rational numbers; and gain a foundation inthe understanding of integers. Students will solve linear equations and use algebraic terminology.Students will solve problems involving area, perimeter, and surface area, work with π (pi), andfocus on the relationships among the properties of quadrilaterals. In addition, students will focuson applications of probability and statistics.While learning mathematics, students will be actively engaged, using concrete materials andappropriate technology such as calculators, computers, and spreadsheets. However, facility in theuse of technology shall not be regarded as a substitute for a student’s understanding ofquantitative concepts and relationships or for proficiency in basic computations. Students willalso identify real-life applications of the mathematical principles they are learning and applythese to science and other disciplines they are studying.Mathematics has its own language, and the acquisition of specialized vocabulary and languagepatterns is crucial to a student’s understanding and appreciation of the subject. Students shouldbe encouraged to use correctly the concepts, skills, symbols, and vocabulary identified in thefollowing set of standards.Problem solving has been integrated throughout the six content strands. The development ofproblem-solving skills should be a major goal of the mathematics program at every grade level.Instruction in the process of problem solving will need to be integrated early and continuouslyinto each student’s mathematics education. Students must be helped to develop a wide range ofskills and strategies for solving a variety of problem types.
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011Number and Number SenseFocus: Relationships among Fractions, Decimals, and Percents6.1 The student will describe and compare data, using ratios, and will use appropriate a notations, such as , a to b, and a:b. b6.2 The student will a) investigate and describe fractions, decimals, and percents as ratios; b) identify a given fraction, decimal, or percent from a representation; c) demonstrate equivalent relationships among fractions, decimals, and percents; and d) compare and order fractions, decimals, and percents.6.3 The student will a) identify and represent integers; b) order and compare integers; and c) identify and describe absolute value of integers.6.4 The student will demonstrate multiple representations of multiplication and division of fractions.6.5 The student will investigate and describe concepts of positive exponents and perfect squares.Computation and EstimationFocus: Applications of Operations with Rational Numbers6.6 The student will a) multiply and divide fractions and mixed numbers; and b) estimate solutions and then solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.6.7 The student will solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals.6.8 The student will evaluate whole number numerical expressions, using the order of operations.MeasurementFocus: Problem Solving with Area, Perimeter, Volume, and Surface Area6.9 The student will make ballpark comparisons between measurements in the U.S. Customary System of measurement and measurements in the metric system.6.10 The student will a) define π (pi) as the ratio of the circumference of a circle to its diameter;
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011 b) solve practical problems involving circumference and area of a circle, given the diameter or radius; c) solve practical problems involving area and perimeter; and d) describe and determine the volume and surface area of a rectangular prism.GeometryFocus: Properties and Relationships6.11 The student will a) identify the coordinates of a point in a coordinate plane; and b) graph ordered pairs in a coordinate plane.6.12 The student will determine congruence of segments, angles, and polygons.6.13 The student will describe and identify properties of quadrilaterals.Probability and StatisticsFocus: Practical Applications of Statistics6.14 The student, given a problem situation, will a) construct circle graphs; b) draw conclusions and make predictions, using circle graphs; and c) compare and contrast graphs that present information from the same data set.6.15 The student will a) describe mean as balance point; and b) decide which measure of center is appropriate for a given purpose.6.16 The student will a) compare and contrast dependent and independent events; and b) determine probabilities for dependent and independent events.Patterns, Functions, and AlgebraFocus: Variable Equations and Properties6.17 The student will identify and extend geometric and arithmetic sequences.6.18 The student will solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions.6.19 The student will investigate and recognize a) the identity properties for addition and multiplication; b) the multiplicative property of zero; and c) the inverse property for multiplication.6.20 The student will graph inequalities on a number line.
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011Grade SevenThe seventh-grade standards continue to emphasize the foundations of algebra. Students whosuccessfully complete the seventh-grade standards should be prepared to study Algebra I ingrade eight. Topics in grade seven include proportional reasoning, integer computation, solvingtwo-step linear equations, and recognizing different representations for relationships. Studentswill apply the properties of real numbers in solving equations, solve inequalities, and use dataanalysis techniques to make inferences, conjectures, and predictions.While learning mathematics, students will be actively engaged, using concrete materials andappropriate technology such as calculators, computers, and spreadsheets. However, facility in theuse of technology shall not be regarded as a substitute for a student’s understanding ofquantitative concepts and relationships or for proficiency in basic computations. Students willalso identify real-life applications of the mathematical principles they are learning and applythese to science and other disciplines they are studying.Mathematics has its own language, and the acquisition of specialized vocabulary and languagepatterns is crucial to a student’s understanding and appreciation of the subject. Students shouldbe encouraged to use correctly the concepts, skills, symbols, and vocabulary identified in thefollowing set of standards.Problem solving has been integrated throughout the six content strands. The development ofproblem-solving skills should be a major goal of the mathematics program at every grade level.Instruction in the process of problem solving will need to be integrated early and continuouslyinto each student’s mathematics education. Students must be helped to develop a wide range ofskills and strategies for solving a variety of problem types.Number and Number SenseFocus: Proportional Reasoning7.1 The student will a) investigate and describe the concept of negative exponents for powers of ten; b) determine scientific notation for numbers greater than zero; c) compare and order fractions, decimals, percents, and numbers written in scientific notation; d) determine square roots; and e) identify and describe absolute value for rational numbers.7.2 The student will describe and represent arithmetic and geometric sequences, using variable expressions.Computation and EstimationFocus: Integer Operations and Proportional Reasoning
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 20117.3 The student will a) model addition, subtraction, multiplication, and division of integers; and b) add, subtract, multiply, and divide integers.7.4 The student will solve single-step and multistep practical problems, using proportional reasoning.MeasurementFocus: Proportional Reasoning7.5 The student will a) describe volume and surface area of cylinders; b) solve practical problems involving the volume and surface area of rectangular prisms and cylinders; and c) describe how changing one measured attribute of a rectangular prism affects its volume and surface area.7.6 The student will determine whether plane figures—quadrilaterals and triangles—are similar and write proportions to express the relationships between corresponding sides of similar figures.GeometryFocus: Relationships between Figures7.7 The student will compare and contrast the following quadrilaterals based on properties: parallelogram, rectangle, square, rhombus, and trapezoid.7.8 The student, given a polygon in the coordinate plane, will represent transformations (reflections, dilations, rotations, and translations) by graphing in the coordinate plane.Probability and StatisticsFocus: Applications of Statistics and Probability7.9 The student will investigate and describe the difference between the experimental probability and theoretical probability of an event.7.10 The student will determine the probability of compound events, using the Fundamental (Basic) Counting Principle.7.11 The student, given data for a practical situation, will a) construct and analyze histograms; and b) compare and contrast histograms with other types of graphs presenting information from the same data set.
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011Patterns, Functions, and AlgebraFocus: Linear Equations7.12 The student will represent relationships with tables, graphs, rules, and words.7.13 The student will a) write verbal expressions as algebraic expressions and sentences as equations and vice versa; and b) evaluate algebraic expressions for given replacement values of the variables.7.14 The student will a) solve one- and two-step linear equations in one variable; and b) solve practical problems requiring the solution of one- and two-step linear equations.7.15 The student will a) solve one-step inequalities in one variable; and b) graph solutions to inequalities on the number line.7.16 The student will apply the following properties of operations with real numbers: a) the commutative and associative properties for addition and multiplication; b) the distributive property; c) the additive and multiplicative identity properties; d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero.Grade EightThe eighth-grade standards are intended to serve two purposes. First, the standards containcontent that reviews or extends concepts and skills learned in previous grades. Second, theycontain new content that prepares students for more abstract concepts in algebra and geometry.The eighth-grade standards provide students additional instruction and time to acquire theconcepts and skills necessary for success in Algebra I. Students will gain proficiency incomputation with rational numbers and will use proportions to solve a variety of problems. Newconcepts include solving multistep equations and inequalities, graphing linear equations,visualizing three-dimensional shapes represented in two-dimensional drawings, and applyingtransformations to geometric shapes in the coordinate plane. Students will verify and apply thePythagorean Theorem and represent relations and functions, using tables, graphs, and rules. Theeighth-grade standards provide a more solid foundation in Algebra I for those students not readyfor Algebra I in grade eight.While learning mathematics, students will be actively engaged, using concrete materials andappropriate technologies. However, facility in the use of technology shall not be regarded as asubstitute for a student’s understanding of quantitative concepts and relationships or forproficiency in basic computations. Students will also identify real-life applications of themathematical principles they are learning that can be applied to science and other disciplinesthey are studying.
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011Mathematics has its own language, and the acquisition of specialized vocabulary and languagepatterns is crucial to a student’s understanding and appreciation of the subject. Students shouldbe encouraged to use correctly the concepts, skills, symbols, and vocabulary identified in thefollowing set of standards.Problem solving has been integrated throughout the six content strands. The development ofproblem-solving skills should be a major goal of the mathematics program at every grade level.Instruction in the process of problem solving will need to be integrated early and continuouslyinto each student’s mathematics education. Students must be helped to develop a wide range ofskills and strategies for solving a variety of problem types.Number and Number SenseFocus: Relationships within the Real Number System8.1 The student will a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers; and b) compare and order decimals, fractions, percents, and numbers written in scientific notation.8.2 The student will describe orally and in writing the relationships between the subsets of the real number system.Computation and EstimationFocus: Practical Applications of Operations with Real Numbers8.3 The student will a) solve practical problems involving rational numbers, percents, ratios, and proportions; and b) determine the percent increase or decrease for a given situation.8.4 The student will apply the order of operations to evaluate algebraic expressions for given replacement values of the variables.8.5 The student will a) determine whether a given number is a perfect square; and b) find the two consecutive whole numbers between which a square root lies.MeasurementFocus: Problem Solving8.6 The student will a) verify by measuring and describe the relationships among vertical angles, adjacent angles, supplementary angles, and complementary angles; and
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011 b) measure angles of less than 360°.8.7 The student will a) investigate and solve practical problems involving volume and surface area of prisms, cylinders, cones, and pyramids; and b) describe how changing one measured attribute of a figure affects the volume and surface area.GeometryFocus: Problem Solving with 2- and 3-Dimensional Figures8.8 The student will a) apply transformations to plane figures; and b) identify applications of transformations.8.9 The student will construct a three-dimensional model, given the top or bottom, side, and front views.8.10 The student will a) verify the Pythagorean Theorem; and b) apply the Pythagorean Theorem.8.11 The student will solve practical area and perimeter problems involving composite plane figures.Probability and StatisticsFocus: Statistical Analysis of Graphs and Problem Situations8.12 The student will determine the probability of independent and dependent events with and without replacement.8.13 The student will a) make comparisons, predictions, and inferences, using information displayed in graphs; and b) construct and analyze scatterplots.Patterns, Functions, and AlgebraFocus: Linear Relationships8.14 The student will make connections between any two representations (tables, graphs, words, and rules) of a given relationship.8.15 The student will a) solve multistep linear equations in one variable with the variable on one and two sides of the equation;
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011 b) solve two-step linear inequalities and graph the results on a number line; and c) identify properties of operations used to solve an equation.8.16 The student will graph a linear equation in two variables.8.17 The student will identify the domain, range, independent variable, or dependent variable in a given situation. The Common Core Standards for Mathematics http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf The Common Core Standards Key Points in Mathematics● The K-5 standards provide students with a solid foundation in whole numbers, addition, subtraction, multiplication, division, fractions and decimals—which help young students build the foundation to successfully apply more demanding math concepts and procedures, and move into applications.● In kindergarten, the standards follow successful international models and recommendations from the National Research Council’s Early Math Panel report, by focusing kindergarten work on the number core: learning how numbers correspond to quantities, and learning how to put numbers together and take them apart (the beginnings of addition and subtraction).● The K-5 standards build on the best state standards to provide detailed guidance to teachers on how to navigate their way through knotty topics such as fractions, negative numbers, and geometry, and do so by maintaining a continuous progression from grade to grade.● The standards stress not only procedural skill but also conceptual understanding, to make sure students are learning and absorbing the critical information they need to succeed at higher levels - rather than the current practices by which many students learn enough to get by on the next test, but forget it shortly thereafter, only to review again the following year.● Having built a strong foundation K-5, students can do hands on learning in geometry, algebra and probability and statistics. Students who have completed 7th grade and mastered the content and skills through the 7th grade will be well-prepared for algebra in grade 8.● The middle school standards are robust and provide a coherent and rich preparation for high school mathematics.● The high school standards call on students to practice applying mathematical ways of thinking to real world issues and challenges; they prepare students to think and reason mathematically.
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CRIN E07: Elementary Math Curriculum and InstructionDr. Margie Mason/Ms. Rachael CoferNovember 7, 2011● The high school standards set a rigorous definition of college and career readiness, by helping students develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly do.● The high school standards emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions. For example, the draft standards state: “Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. It is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social and everyday situations can be modeled using mathematical and statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data.” ResourcesGewertz, C. (2011). Experts: Educators cant separate common core state standards. Education Week, 30(36), 24.Kepner Jr., H.S. (2010). A math perspective on the common core standards initiative. NSTA Reports!, 3-4.Wiggins, G. (2011). The common-core math standards: They don’t add up. Education Week, 31 (5), 22-23.http://www.doe.virginia.gov/VDOE/Superintendent/Sols/home.shtmlhttp://www.corestandards.org/the-standardshttp://www.corestandards.org/about-the-standards/key-points-in-mathematics
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