Rapid City Area Schools

Transcription

1 Rapid City Area Schools MIDDLE SCHOOL Mathematics Curriculum Including Algebra 1 Approved by the Boards of Education, November 2002 Revision approved by the Board of Education, June 2005

2 Secondary Mathematics Curriculum Re-alignment Committee Name School Grade Middle School members Donna Greenwalt Southwest Middle Seventh Kirk Guymon Southwest Middle Eighth Sally Heberlein Secondary Math Teacher Leader Rosemary Johnson West Middle Sixth Rebecca Kline North Middle Sixth Bailey Kowalski Dakota Middle Eighth Stacy Krumpus Dakota Middle Eighth Ken Kundel North Middle Sixth Jill Trainer West Middle Eighth High School members Jeff Barnes Stevens High Dan Conrad Central High Ruth Conway Stevens High Al Johnson Stevens High Tom Keck Stevens High Marie Ritten Secondary Math Coordinator Our thanks to Josh Lien, 6 th grade math teacher at Southwest Middle School for creating, formatting, and offering to add the South Dakota Abbreviated Standards pages to this document.

3 RCAS Middle School Curriculum TABLE OF CONTENTS INTRODUCTION Introduction...ii Philosophy iii Essential Knowledge. iv NCTM Principles v NCTM Content Standards 2000 vi NCTM Process Standards 2000 vii New MIDDLE SCHOOL The following content standards & curricula are arranged alphabetically by math strand. This does not in any way determine the order of importance or implementation. Vertical Alignment Tables...1 RCAS Grades 6-8 Curriculums..12 SD Content Standards/RCAS Curriculum 6 th grade th grade th grade.. 36 Algebra I Curriculum.. 42 SD Abbreviated Math Standards with examples 6 th grade th grade th grade.. 53 South Dakota Standards Glossary.. 56 New

4 Introduction

5 Rapid City Area Schools Math Curriculum 2005 This document was written by the members of the Math Vertical Alignment Team to reflect the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics 2000 and South Dakota Standards for Mathematics Instruction. Marie Ritten, Secondary Math Coordinator, and Sally Heberlein, Secondary Math Teacher Leader, facilitated the Math Vertical Alignment Team committee. The following document was approved by the Instructional Council and authorized by the Board of Education for the Rapid City Area School (RCAS). Vision Statement Our vision is the development of a curriculum with common language and continuity. It provides a strong and diverse foundation where the teaching and learning of mathematics are in harmony to meet individual needs that promote success, confidence, and an insatiable thirst for learning.

6 Introduction The need to understand and be able to use mathematics in everyday life and in the workplace has never been greater and will continue to increase. In this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures. Mathematical competence opens doors to productive futures. The Math Vertical Alignment Committee was formed in 2004 and charged with the task of reviewing and revising the RCAS Secondary Mathematics Curriculum of One major objective in this process was the vertical alignment of the Rapid City Math Curriculum for grades Another major objective for this committee was the alignment of the Rapid City Math Curriculum with the South Dakota Mathematics Content Standards 2004 and the NCTM Principles and Standards for School Mathematics This curriculum provides Rapid City students with rigorous topics beyond those of the South Dakota Mathematics Content Standards In order to assure proficiency for each student the standards have been categorized as Mastery, Develop, and Introduce for each of the grade levels. Cleary there is more to teaching and learning than these standards. Adjustments will need to be made for those students who exceed the standards and for those who cannot easily meet them. The standards are a starting point in creating an environment where all students can learn to live and thrive in a constantly changing and increasingly complex world. Page ii

7 Page iii Grades Pre-K 12 Philosophy The educational mission of the Rapid City area Schools is to prepare students to lead personal fulfilling and responsible lives. For its part, mathematics education should provide students with a useful base of mathematical knowledge and skills that will enable them to think and reason mathematically. The Mathematics Curriculum is designed to develop mathematical reasoning skills, conceptual understanding of mathematics, and mathematically powerful students who: value mathematics. are confident and proficient in their ability to do mathematics. are mathematical problem-solvers. can communicate mathematically. reason mathematically. Our students deserve and need the best mathematics education possible, one that enables them to fulfill personal ambitions and career goals in an ever-changing world. The NCTM Principles and Standards for School Mathematics describe particular features of high-quality mathematics.

8 Essential Knowledge The RCAS Mathematics Curriculum is based on the South Dakota Mathematics Content Standards of Grade-level standards specify what students should know and be able to do by the end of each grade level. While curriculum specifies what teachers will teach. Since standards are not the same as curriculum, the review topics do not appear from grade to grade across grade-level standards. Teachers from prior years are charged with introducing and developing skills before grade-level mastery is expected. Teachers and researchers have learned that in order for students to demonstrate mastery of skills specified in the standards on summative (end-of-year) assessments, teachers must teach, and students must learn at a level of fluency that exceeds the apparent expectations of the grade-level standard. Goals: The five broad, conceptual goals (content area/discipline standards) are the K-12 strands, which are the common threads that represent expected outcomes for all students preparing to graduate from South Dakota schools. Because the goals are the end results of what we would expect after thirteen years of mathematics study, they are worded the same at each grade level. This done to provide consistency in K-12 curricular focus and alignment. Indicators: The indicators further define the goals and set the framework for mathematics. The indicators remain the same at all instructional levels (K-2, 3-5, 6-8, 9-12), thereby providing an ongoing and constant focus for the standards. The indicators also provide the targets and anchors for broad district-level, program evaluation. Grade Level Standards: These statements represent the classroom learning objectives or activities which should be provided at each grade level to help students reach the expectations articulated in the indicators, and goals. Page iv

9 Page v NCTM Principles Equity: Excellence in mathematics education requires equity---high expectations and strong support for all students. Curriculum: A curriculum is more that a collection of activities; it must be coherent, focused on important mathematics and well articulated across the grades. Teaching: Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. Learning: Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Assessment: Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. Technology: Technology is essential in teacher and learning mathematics; it influences the mathematics that is taught and enhances students learning.

10 NCTM Content Standards Instructional programs from pre-kindergarten through grade 12 should enable all students to: Number and Operations Standard understand numbers, ways of representing numbers, relationships among numbers, and number systems. understand meanings of operations and how they relate to one another. compute fluently and make reasonable estimates. Algebra Standard with Patterns, Relations understand patterns, relations, and functions. represent and analyze mathematical situations and structures using algebraic symbols. use mathematical models to represent and understand quantitative relationships. analyze change in various contexts. Geometry Standard analyze characteristics and properties of twoand three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. specify locations and describe spatial relationships using coordinate geometry and other representational systems. apply transformations and use symmetry to analyze mathematical situations. use visualization, spatial reasoning, and geometric modeling to solve problems. Measurement Standard understand measurable attributes of objects and the units, systems, and processes of measurement. apply appropriate techniques, tools, and formulas to determine measurements. Data Analysis and Probability Standard formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. select and use appropriate statistical methods to analyze data. develop and evaluate inferences and predictions that are based on data. understand and apply basic concepts of probability. Page vi

11 Page vii NCTM Process Standards Instructional programs from pre-kindergarten through grade 12 should enable all students to: Problem Solving Standard build new mathematical knowledge through problem solving. solve problems that arise in mathematics and in other contexts. apply and adapt a variety of appropriate strategies to solve problems. monitor and reflect on the process of mathematical problem solving. Reasoning and Proof Standard recognize reasoning and proof as fundamental aspects of mathematics. make and investigate mathematical conjectures. develop and evaluate mathematical arguments and proofs. select and use various types of reasoning and methods of proof. Communication Standard organize and consolidate their mathematical thinking through communication. communicate their mathematical thinking coherently and clearly to peers, teachers, and others. analyze and evaluate the mathematical thinking and strategies of others. use the language of mathematics to express mathematical ideas precisely. Connections Standard recognize and use connections among mathematical ideas. understand how mathematical idea interconnect and build on one another to produce a coherent whole. recognize and apply mathematics in contexts outside of mathematics. Representation Standard create and use representations to organize, record, and communicate mathematical ideas. select, apply, and translate among mathematical representations to solve problems. use representations to model and interpret physical, social, and mathematical phenomena.

12 Vertical Alignment Tables

13 Key to abbreviations I Introduce D- Develop M- Master South Dakota Content Standards ALGEBRA 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 SIXTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D M 6.A.1.1. (Application) Use order of operations, excluding nested parentheses and exponents, to simplify whole number expressions. I D M 6.A.1.2. (Application) Write algebraic expressions involving addition or multiplication using whole numbers. I D M 6.A.2.1. (Application) Write and solve one-step 1 st degree equations, with one variable, involving inverse operations using the set of whole numbers. SEVENTH GRADE I D/M 6.A.3.1. (Knowledge) Identify and graph ordered pairs in Quadrant I on a coordinate I/D M 6.A.3.2. (Application) Solve one-step problems involving ratios and rates. plane. 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M 7.A.1.1. (Application) Write and evaluate algebraic expressions using the set of whole numbers. I D D M 7.A.1.2. (Knowledge) Identify associative, commutative, distributive, and identify properties involving algebraic expressions. I D D M 7.A.2.1. (Application) Write and solve one-step 1 st degree equations, with one variable, using the set of integers and inequalities, with one variable, using the set of whole numbers. I/D M 7.A.3.1. (Application) Identify and graph ordered pairs on a coordinate plane and inequalities on a number line. I I D M 7.A.3.2. (Application) Model and solve multi-step problems involving rates. I D D M 7.A.4.1. (Application) Recognize one-step patterns using tables, graphs, and models and create one-step algebraic expressions representing the pattern.

14 Page 2 Key to abbreviations I Introduce D- Develop M- Master South Dakota Content Standards ALGEBRA- Continued 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 EIGHTH GRADE/ADVANCED SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I I D D M M 8.A.1.1. (Application) Use properties to expand, combine, and simplify 1 st degree algebraic expressions with the set of integers. I D D D M M 8.A.2.1. (Application) Write and solve two-step 1 st degree equations, with one variable, and one-step inequalities, with one variable, using the set of integers. I D M M 8.A.3.1. (Comprehension) Describe and determine linear relationships. I D M M 8.A.4.1. (Synthesis) Create rules to explain the relationship between numbers when a change in the first variable affects the second variable. I D M M 8.A.4.2. (Analysis) Describe and represent relations using tables, graphs, and rules. ALGEBRA 1 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M 9-12.A.1.1. (Comprehension) Write equivalent forms of algebraic expressions using properties of the set of real numbers. I D D D M 9-12.A.2.1. (Comprehension) Use algebraic properties to transform multi-step, single variable, and first-degree equations. I D D M 9-12.A.2.2. (Application) Use algebraic properties to transform multi-step, single variable, and first-degree inequalities and represent solutions using a number line. I D D D M 9-12.A.3.1. (Application) Create linear models to represent problem situations. I D D M 9-12.A.3.2. (Comprehension) Distinguish between linear and nonlinear models. I D D M 9-12.A.4.1. (Application) Use graphs, tables, and equations to represent linear equations.

15 Key to abbreviations I Introduce D- Develop M- Master South Dakota Content Standards ALGEBRA- Continued 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 ALGEBRA 1 (continued) I 9-12.A.1.1A. (Application) Write equivalent forms of rational algebraic expressions using properties of real numbers. I I I 9-12.A.1.2A. (Application) Extend the use of real number properties to expressions involving complex numbers. I 9-12.A.2.1A. (Analysis) Determine solutions of quadratic equations. I 9-12.A.2.2A. (Application) Determine the solution of systems of equations and systems of inequalities. I I D 9-12.A.2.3A. (Application) Determine solutions to absolute value statements. I I D 9-12.A.3.1A. (Analysis) Distinguish between linear, quadratic, inverse variations, and exponential models. I I D 9-12.A.3.2A. (Synthesis) Create formulas to model relationships that are algebraic, geometric, trigonometric, and exponential. I 9-12.A.3.3A. (Analysis) Use sequences and series to model relationships. I I D 9-12.A.4.1A. (Analysis) Determine the domain, range, and intercepts of a function. I 9-12.A.4.2A. (Analysis) Describe the behavior of a polynomial, given the leading coefficient, roots, and degree.

16 Page 4 South Dakota Content Standards GEOMETRY Key to abbreviations I Introduce D- Develop M- Master 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 SIXTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D M 6.G.1.1. (Comprehension) Identify and describe the characteristics of triangles and quadrilaterals. I D M 6.G.1.2. (Comprehension) Identify and describe angles. I D M 6.G.2.1. (Application) Use basic shapes to demonstrate geometric concepts. SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M 7.G.1.1. (Application) Identify, describe, and classify polygons having up to 10 sides. I D D M 7.G.1.2. (Knowledge) Identify and describe elements of geometric figures. I D D M 7.G.2.1. (Application) Demonstrate ways that shapes can be transformed. EIGHTH GRADE/ADVANCED SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M M 8.G.1.1. (Application) Describe and classify prisms, pyramids, cylinders, and cones. I D D M M 8.G.1.2. (Application) Given any two sides of an illustrated right triangle, use the Pythagorean Theorem to find the third side. I D D M M 8.G.2.1. (Application) Write and solve proportions that express the relationships between corresponding parts of similar quadrilaterals and triangles.

17 South Dakota Content Standards GEOMETRY- Continued Key to abbreviations I Introduce D- Develop M- Master 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 ALGEBRA 1 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D D M 9-12.G.1.1. (Application) Apply the properties of triangles and quadrilaterals to find unknown parts. I D D D D D M 9-12.G.1.2. (Application) Identify and apply relationships among triangles. I D M 9-12.G.2.1. (Analysis) Recognize the relationship between a three-dimensional figure and its two-dimensional representation. I D D D D D M 9-12.G.2.2. (Application) Reflect across vertical or horizontal lines, and translate two-dimensional figures. I I D D M 9-12.G.2.3. (Application) Use proportions to solve problems. I 9-12.G.1.1A. (Evaluation) Justify properties of geometric figures. I 9-12.G.1.2A. (Application) Determine the values of sine, cosine, and tangent ratios of right triangles. I 9-12.G.1.3A. (Application) Apply properties associated with circles. I 9-12.G.1.4A. (Analysis) Use formulas for surface area and volume to solve problems involving three-dimensional figures. I 9-12.G.2.1A. (Synthesis) Use Cartesian coordinates to verify geometric properties. Page 5

18 Page 6 South Dakota Content Standards MEASUREMENT Key to abbreviations I Introduce D- Develop M- Master 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 SIXTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D M 6.M.1.1. (Comprehension) Select, use, and convert appropriate unit of measurement for a situation. I D M 6.M.1.2. (Comprehension) Find the perimeter and area of squares and rectangles (whole number measurements). SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M 7.M.1.1. (Comprehension) Select, use, and convert appropriate unit of measurement for a situation including capacity and angle measurement. I D D M 7.M.1.2. (Comprehension) Given the formulas, find the circumference, perimeter, and area of circles, parallelograms, triangles, and trapezoids (whole number measurement). EIGHTH GRADE/ADVANCED SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D M M 8.M.1.1. (Application) Apply proportional reasoning to solve measurement problems with rational number measurements. I D M M 8.M.1.2. (Comprehension) Find area, volume, and surface area with whole number measurement.

19 South Dakota Content Standards MEASUREMENT - Continued Key to abbreviations I Introduce D- Develop M- Master 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 ALGEBRA 1 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D D M 9-12.M.1.1. (Comprehension) Choose appropriate unit label, scale, and precision. I D D D M 9-12.M.1.2. (Comprehension) Use suitable units when describing rate of change. I D D D D M 9-12.M.1.3. (Application) Use formulas to find perimeter, circumference, and area to solve problems involving common geometric figures. I I D 9-12.M.1.1A. (Application) Use dimensional analysis to check answers and determine units of a problem solution. I I/D I/D D 9-12.M.1.2A. (Analysis) Use indirect measurement in problem situations that defy direct measurement. Page 7

20 Key to abbreviations I Introduce D- Develop M- Master South Dakota Content Standards NUMBER SENSE Page 8 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 SIXTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D M 6.N.1.1. (Comprehension) Represent fractions in equivalent forms and convert between fractions, decimals, and percents using halves, fourths, tenths, and hundredths. I D M 6.N.1.2. (Knowledge) Find factors and multiples of whole numbers. I D M 6.N.2.1. (Comprehension) Add, subtract, multiply, and divide decimals. I D M 6.N.3.1. (Application) Use various strategies to solve one- and two-step problems involving positive decimals. SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M 7.N.1.1. (Comprehension) Represent numbers in a variety of forms by describing, ordering, and comparing integers, decimals, percents, and fractions. I D D M 7.N.1.2. (Application) Find and use common multiples and factors of whole numbers. I I/D D M 7.N.2.1. (Application) Add, subtract, multiply, and divide integers and positive fractions. I D D M 7.N.3.1. (Application) Use various strategies to solve one- and two-step problems involving positive fractions and integers.

21 South Dakota Content Standards NUMBER SENSE- Continued Key to abbreviations I Introduce D- Develop M- Master 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 EIGHTH GRADE/ADVANCED SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M M 8.N.1.1. (Comprehension) Represent numbers in a variety of forms and identify the subsets of rational numbers. I I D M M 8.N.2.1. (Application) Read, write, and compute within any subset of rational numbers. I D D M M 8.N.3.1. (Application) Use various strategies to solve multi-step problems involving rational numbers. ALGEBRA 1 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D D D D M 9-12.N.1.1. (Comprehension) Identify multiple representations of a real number. I D D D D D M 9-12.N.1.2. (Comprehension) Apply the concept of place value, magnitude, and relative magnitude of real numbers. I D D D D D M 9-12.N.2.1. (Comprehension) Add, subtract, multiply, and divide real numbers including integral (integers) exponents. I D D D D D M 9-12.N.3.1. (Analysis) Use estimation strategies in problem situations to predict results and to check the reasonableness of results. I D D D D M 9-12.N.3.2. (Comprehension) Select alternative computational strategies and explain the chosen strategy. I 9-12.N.1.1A. (Comprehension) Describe the relationship of the real number system to the complex number system. I I I 9-12.N.1.2A. (Application) Apply properties and axioms of the real number system to various subsets, e.g., axioms of order, closure. I 9-12.N.2.1A. (Application) Add, subtract, multiply, and divide real numbers including rational exponents. Page 9

22 South Dakota Content Standards STATISTICS & PROBABILITY STANDARDS Page 10 Key to abbreviations I Introduce D- Develop M- Master 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 SIXTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL D D M 6.S.1.1. (Comprehension) Find the mean, mode, and range of an ordered set of positive data. D D M 6.S.1.2. (Application) Display data using bar and line graphs and draw conclusions from data displayed in a graph. D D M 6.S.2.1. (Knowledge) Find the probability of a simple event. SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M 7.S.1.1. (Comprehension) Find the mean, median, mode, and range of a set of data. I D D M 7.S.1.2. (Application) Display data, using frequency tables, line plots, stem-and-leaf plots and make predictions from data displayed in a graph. I D D M 7.S.2.1. (Comprehension) Given a sample space, find the probability of a specific outcome. EIGHTH GRADE/ADVANCED SEVENTH GRADE 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D M M 8.S.1.1. (Comprehension) Find the mean, median, mode, and range of a data set from a stem-and-leaf plot and a line plot. I D D M M 8.S.1.2. (Application) Use a variety of visual representations to display data to make comparisons and predictions.

23 I D D M M 8.S.2.1. (Comprehension) Find the sample space and compute probability for two simultaneous independent events. South Dakota Content Standards STATISTICS & PROBABILITY STANDARDS- Continued Key to abbreviations I Introduce D- Develop M- Master 4 th Fourth Grade Math, 5 th Fifth Grade Math, 6 th Sixth Grade Math, 7 th Seventh Grade Math, 7A Advanced Seventh Grade Math, 8 th Eighth Grade Math, 8A/9 Algebra 1 ALGEBRA 1 4 th 5 th 6 th 7 th 7A 8 th 8A/9 THE STUDENT WILL I D D D M 9-12.S.1.1. (Analysis) Draw conclusions from a set of data. I I/D D M 9-12.S.1.2. (Comprehension) Compare multiple one-variable data sets, using range, interquartile range, mean, mode, and median. I D D D M 9-12.S.1.3. (Analysis) Represent a set of data in a variety of graphical forms and draw conclusions. I 9-12.S.1.2A. (Evaluation) Analyze and evaluate graphical displays of data. I 9-12.S.1.5A. (Application) Use scatter plots, best-fit lines, and correlation coefficients to model and support data. I I/D D M 9-12.S.2.1. (Knowledge) Distinguish between experimental and theoretical probability. I D D D M 9-12.S.2.2. (Comprehension) Predict outcomes of simple events using given theoretical probabilities. I 9-12.S.2.1A. (Application) Use probabilities to solve problems. I 9-12.S.2.2A. (Application) Determine probability of compound, complementary, independent, and mutually exclusive events. Page 11

24 RCAS Curriculums Grades 6-8

25 Page 12 The learner will master: (6.A.1.1) using order of operations to evaluate whole number expressions including parentheses. (6.A.1.2) writing algebraic expressions involving addition and multiplication using whole numbers. (6.A.2.1) writing and solving one-step first degree equations, with one variable, involving inverse operations using the set of whole numbers. (6.A.3.1) identifying and graphing ordered pairs in Quadrant I on a coordinate plane. (6.A.3.2) solving one-step problems involving ratios and rates. (6.A.4.1) using concrete materials, graphs, and algebraic statements to represent problem situations. o recognizing, describing, and extending arithmetic sequences and patterns o using variables to represent given quantities in problem situations RCAS Curriculum Algebra The learner will master: (7.A.1.1) using order of operations to evaluate whole number algebraic expressions using exponents and parentheses. (7.A.1.2) identifying and using the associative, commutative, distributive, and identity properties involving algebraic expressions. (7.A.2.1) writing simple expressions using variables. o converting from word phrases to symbols (7.A.2.1) writing and solving one-step equations and inequalities using integers. (7.A.3.1) identifying and graphing ordered pairs on a coordinate plane. (7.A.3.1) solving and graphing simple one-step inequalities. (7.A.3.2) modeling and solving multistep problems involving rates. o best buy, fastest speed, greatest rate (7.A.4.1) writing algebraic statements to represent patterns observed in real life situations using tables, graphs, and models. The learner will master: (8.A.1.1) using the commutative, associative, distributive, and identity, properties to expand and simplify algebraic expressions with the set of integers. (8.A.1.1) using order of operations with exponents and nested parentheses to simplify algebraic expressions with integers. (8.A.1.1) determining if two 1 st degree algebraic expressions are equivalent. (8.A.2.1) writing and solving two-step equations using the set of integers. (8.A.2.1) writing and solving one-step inequalities using the set of integers. (8.A.3.1) describing and determining linear relationships. o determining slope from a line or ordered pairs on a graph, identifying x- and y-intercepts from a graph (8.A.4.1) explaining how a change in one variable affects the other variable. o when given the equation x + 10 = y or y = 6x, as x increases what happens to y

26 The learner will develop: understanding the significance of the equality and inequality signs. identifying associative, commutative, distributive, and identity properties involving algebraic expressions. identifying and graphing an inequality on a number line. modeling and solving multi-step problems using unit rates. recognizing one-step patterns using tables, graphs, and models. The learner will develop: using the associative, commutative, distributive, and identity properties to expand and simplify algebraic expressions with the set of integers. identifying x- and y-intercepts from a graph. (8.A.4.2) describing and representing relations using tables, graphs and rules. o using function tables, number machines, or input-output tables The learner will develop: writing equivalent forms of algebraic expressions using properties of the set of real numbers. creating linear models to represent problem situations. The learner will be introduced to: evaluating powers and exponents. solving one-step equations using integers. The learner will be introduced to: using order of operations with nested parentheses. writing and solving two-step equations. identifying positive and negative slope. o steepness The learner will be introduced to: distinguishing between linear and nonlinear models. using graphs, tables, and equations to represent linear functions. o create a table from a graph or an equation of a line Page 13

27 The learner will master: (6.G.1.1) identifying, classifying, and describing characteristics of: o triangles by angle acute, obtuse, right o triangles by side lengths scalene, isosceles, equilateral o quadrilaterals by shape parallelogram, rectangle, rhombus, square (6.G.1.2) identifying, classifying, and describing angles by size acute, obtuse, right. (6.G.2.1) using basic shapes to demonstrate geometric concepts of: o symmetry, reflections, congruency, similarity, perpendicular lines, and parallel lines RCAS Curriculum Geometry The learner will master: (7.G.1.1) identifying, describing, and classifying polygons up to ten sides. o identifying relationships among triangles o identifying relationships among quadrilaterals o sketching two-dimensional figures (7.G.1.2) identifying and describing elements of geometric figures altitudes, midpoints, bisectors, radii, diameters, and chords. (7.G.2.1) demonstrating ways that shapes can be transformed (flips/reflections, slides/translations, and turns/rotations). The learner will master: (8.G.1.1) describing and classifying prisms, pyramids, cylinders, and cones (faces, vertices, edges). (8.G.1.1) constructing three-dimensional figures from two-dimensional views using nets. (8.G.1.2) calculating the third side of a right triangle when given two sides using the Pythagorean Theorem (with whole numbers). (8.G.2.1) writing and solving proportions that express relationships between corresponding parts of similar quadrilaterals and triangles. Page 14 The learner will develop: identifying the parts of circle radius and diameter. identifying, describing, and classifying polygons up to ten sides. identifying relationships among triangles. identifying relationships among quadrilaterals. sketching two-dimensional figures. The learner will develop: describing and classifying prisms, pyramids, cylinders, and cones (faces, vertices, edges). calculating the third side of a right triangle when given two sides, using the Pythagorean Theorem (with the set of whole numbers). writing and solving proportions that The learner will develop: using visual perspective to analyze geometric problems (construct threedimensional models given top, side, or bottom views of objects). creating two-dimensional representations that demonstrate various perspectives of three-dimensional objects. recognizing and identifying the relationships between angles created by

28 identifying and describing elements of geometric figures altitudes, midpoints, bisectors, radii, diameters, and chords. demonstrating ways that shapes can be transformed (flips/reflections, slides/translations, and turns/rotations). express relationships between corresponding parts of similar quadrilaterals and triangles. manipulating tessellations. applying the properties of triangles and quadrilaterals to find unknown parts. intersecting lines using the concepts of complementary, supplementary, vertical, and corresponding angles. identifying corresponding parts of congruent and similar triangles and polygons. applying the properties of triangles and quadrilaterals to find unknown parts. The learner will be introduced to: describing and classifying prisms, pyramids, cylinders, and cones (faces, vertices, edges). using proportions to solve problems. applying the properties of triangles and quadrilaterals to find unknown parts. The student will be introduced to: identifying three-dimensional shapes using nets. recognizing and identifying the relationships between angles created by intersecting lines using the concepts of complementary, supplementary, vertical, and corresponding angles. identifying corresponding parts of congruent and similar triangles and polygons. using proportions to solve problems. The learner will be introduced to: using parallel lines cut by a transversal to identify the congruent angles created. Page 15

29 Page 16 The learner will master: (6.M.1.1) selecting, using, and converting appropriate units of measurement for a situation. o determining elapsed time o converting length, capacity, and mass within the Metric System (kilo-, base unit, centi-, milli-) o converting weight and length within U.S. Customary System (6.M.1.2) finding the perimeter and area of squares and rectangles (whole number measurements). o applying strategies and/or formulas o using appropriate units of measure RCAS Curriculum Measurement Standard The learner will master: (7.M.1.1) selecting, using, and converting appropriate units of measurement for a situation including capacity and angle measurement. o measuring angles o measuring length, capacity, and mass o converting within the Metric System (kilo- thru milli-) o converting within the U.S. Customary System (weight, length, capacity) (7.M.1.2) finding circumference, perimeter, and area of circles, triangles, and trapezoids, when given the formulas (whole number measurements). o using appropriate units of measure o estimating the area of irregular shapes The learner will master: (8.M.1.1) applying proportional reasoning to solve measurement problems with rational number measurements. o converting within measurement systems o using scale drawings o calculating indirect measurement (8.M.1.2) finding area, volume, and surface area with rational number measurements. o using appropriate units of measure o applying strategies and/or formulas o calculating volume of rectangular prisms, rectangular pyramids, cylinders, and cones o calculating area and surface area of composite shapes The learner will develop: selecting, using, and converting units of measurement. o measuring angles using a protractor o measuring and converting length, capacity, and mass, using the U.S. The learner will develop: applying proportional reasoning to solve measurement problems with rational number measurements. o converting within measurement systems o using scale drawings The learner will develop: choosing appropriate unit label, scale, and precision. o determining appropriate scales for histograms, scatter plots, and other graphs

30 Customary System or Metric System (kilo- thru milli-) finding circumference, perimeter, and area of circles, triangles, and trapezoids, when given the formulas. o using appropriate units of measure (whole numbers, decimals, and fractions) o estimating the area of irregular shapes The learner will be introduced to: examining measurement situations to determine necessary degree of accuracy. finding sums and differences of various units within a measurement system (yards, feet, inches). using suitable units when describing rate of change. applying proportional reasoning to solve measurement problems with whole number measurements. o using scale drawings to represent situations calculating area, volume, and surface area with whole number measurements. o applying strategies and/or formulas o finding volume of rectangular prisms, rectangular pyramids, cylinders, and cones o calculating area and surface area of composite shapes o calculating indirect measurement finding area, volume, and surface area with whole number measurements. o using appropriate units of measure o applying strategies and/or formulas o calculating area of composite shapes using suitable units when describing rate of change. examining measurement situations to determine necessary degree of accuracy. The learner will be introduced to: using indirect measurement in problem situations that defy direct measurement. choosing appropriate unit label, scale, and precision. o determining appropriate scales for histograms, scatter plots, and other graphs using suitable units when describing rate of change. knowledge about using formulas that find perimeter, circumference, and area to solve problems involving common geometric figures. o using algebraic expressions with geometric formulas using indirect measurement in problem situations that defy direct measurement. The learner will be introduced to: using dimensional analysis to check answers and determine units of a problem solution o converting rate of change between measurement systems o 2 cups * 2 pints * 4 quarts 1 pint 1 quart 1 gallon = 16 cups 1 gallon Page 17

31 Page 18 The learner will master: (6.N.1.1) comparing and ordering decimal numbers to the ten-thousandths place. (6.N.1.1) representing fractions in equivalent forms and converting between fractions, decimals, and percents using halves, fourths, tenths, and hundredths. (6.N.1.1) converting between mixed numbers and improper fractions. o identifying both standard and word forms (millions to ten-thousandths) of positive rational numbers (6.N.1.2) finding factors (including prime factorization) and multiples of whole numbers. (6.N.1.2) classifying numbers as prime or composite. o using the divisibility rules for 2, 3, 5, 6, 9, & 10 (6.N.2.1) adding, subtracting, multiplying, and dividing numbers in decimal form to the tenthousandths place. (6.N.3.1) using various strategies to solve one and two step problems involving positive decimals. o rounding off numbers in decimal form to the ten-thousandths place o formulating rules to solve practical problems. RCAS Curriculum Number Sense Standard The learner will master: (7.N.1.1) representing numbers in a variety of forms by describing, ordering and comparing integers, decimals, fractions, and percents. o identifying and representing terminating and repeating decimals (7.N.1.1) determining whether ratios are equal and proportional. (7.N.1.2) finding and using common factors and multiples of whole numbers. o calculating the GCF and LCM using prime factorization o using the divisibility rules for 2, 3, 4, 5, 6, 8, 9, and 10 (7.N.2.1) adding, subtracting, multiplying, and dividing integers and positive fractions with like and unlike denominators. (7.N.3.1) using various strategies to solve one and two step problems involving positive fractions and integers. The learner will master: (8.N.1.1) representing numbers in a variety of forms (radicals, absolute value, exponents, and scientific notation). (8.N.1.1) identifying the subsets of rational numbers (counting/natural, whole, integers). (8.N.2.1) reading, writing and computing within any subset of rational numbers. (8.N.2.1) solving consumer application problems involving discounts, mark-ups, commissions, profits, and simple interest. (8.N.3.1) using various strategies to solve multi-step problems involving rational numbers. (8.N.3.1) formulating rules to solve practical problems involving rational numbers. (8.N.3.1) using estimation strategies to make predictions and test the reasonableness of the answer. (8.N.3.1) simplifying expressions using properties to justify steps.

32 The learner will develop: finding the least common denominator of two or more fractions. expressing fractions in simplest form. adding, subtracting, multiplying, and dividing positive, proper fractions and mixed numbers with like denominators. comparing and ordering integers. modeling, adding, and subtracting integers. understanding the meaning and effects of operations on rational numbers. The learner will develop: finding the absolute value of integers. representing numbers in a variety of equivalent forms (positive and negative exponents, scientific notation, absolute value). developing the meaning of percents less than one and greater than one hundred. identifying, representing, comparing and ordering rational numbers and representing them on a number line. solving consumer application problems involving discounts, mark-ups, commissions, profits, and simple compound interest. using estimation strategies to make predictions and test the reasonableness of the answer. formulating rules to solve practical problems involving rational numbers. using proportions to solve a variety of problems. using estimation strategies to make predictions and test the reasonableness of the answer. The learner will develop: solving inequalities that require multiplication and division by a negative number. comparing and order radicals, numbers expressed with exponents, and the absolute value of various numbers. using multiple representations of a given real number. The learner will be introduced to: adding, subtracting, multiplying, and dividing positive, proper fractions and mixed numbers with unlike denominators. solving consumer application problems involving discounts, mark-ups, commissions, profits, and simple compound interest. representing numbers in scientific notation. using estimation strategies to test the reasonableness of the answer. The learner will be introduced to: using mathematical symbols to represent irrational numbers (i.e. square root symbol,, and bar notation). comparing and ordering radicals, numbers expressed with exponents, and the absolute value of various numbers. The learner will be introduced to: identifying and classifying irrational numbers in the real number system and finding their approximate Pagelocation 20 on the number line. identifying the various subsets of the real number system (rational and irrational numbers) Page 19

33 using estimation strategies to make predictions and test the reasonableness of the answer.

34 The learner will master: (6.S.1.1) finding the mean, median, mode, and range of an ordered set of positive data. (6.S.1.2) reading and drawing conclusions from tables and graphs. (6.S.1.2) solving problems by organizing data into tables and graphs. (6.S.1.2) constructing bar and line graphs using data. (6.S.2.1) recognizing the probability of a simple event. o coin flipping, single die rolling, spinner card RCAS Curriculum Statistics and Probability The learner will master: (7.S.1.1) finding the mean, median, mode, and range of an unordered set of data. (7.S.1.2) interpreting and constructing bar graphs, line plots, circle graphs, and stem-andleaf plots using appropriate scales, labels, and intervals. (7.S.1.2) solving and analyzing problems by organizing data into tables and frequency charts. (7.S.1.2) using a variety of visual representations to make predictions (bar graphs, line plots, circle graphs, stem-and-leaf plots). (7.S.2.1) calculating probability, given a sample space, with a known number of outcomes. o possible outcome from flipping two coins simultaneously The learner will master: (8.S.1.1) systematically collecting, organizing, describing, and analyzing sets of data to make simple predictions using mean, median, mode, and range on stemand-leaf plots and line plots. (8.S.1.2) using a variety of visual representations to display data to make comparisons and predictions (i.e. doublebar graphs, double-line graphs, and scatter plots). (8.S.2.1) finding the sample space and compute probability of two simultaneous, independent events. The learner will develop: using a variety of visual representations to make predictions (bar graph, line plots, circle graphs, stem-and-leaf plots). expressing probability as a ratio, decimal, and percent. making predictions based on the probability of an independent event. calculating the sample space of an independent event. The learner will develop: plotting ordered pairs on a scatter plot. computing the probability of two simultaneous, independent events. The learner will develop: identifying the line of best fit (line of regression) on a scatter plot to determine the type of correlation (positive, negative, or no correlation). understanding the difference between experimental and theoretical probabilities. using various ways to calculate the number of possible outcomes (The Fundamental Counting Principle).

35 o possible outcome from flipping two coins simultaneously understanding the concept of sampling bias and ways to select unbiased samples. The learner will be introduced to: interpreting circle graphs and stem-and-leaf plots. using the most appropriate use of central tendency. plotting ordered pairs on a scatter plot. organizing data into frequency tables. computing the probability for two simultaneous, independent events. The learner will be introduced to: using central tendency in real world situations. identifying the line of best fit (line of regression) on a scatter plot to determine the type of correlation. recognizing the difference between experimental and theoretical probabilities. using various ways to calculate the number of possible outcomes (The Fundamental Counting Principle). understanding the concept of sampling bias and ways to select unbiased samples. The learner will introduced to: exploring the interquartile range using boxand-whisker plots. calculating the probability of a compound event. Page 21

36 SD Content Standards/ RCAS Curriculum This section combines the SD Mathematics Content Standards of 2004 and the realigned RCAS Curriculum. It was designed with the teacher in mind, and its main purpose is to aid in the planning of mathematics lessons for the students of RCAS. The layout is as follows: Indicator #: K 12 indicator The indicators further define the goals and set the framework for mathematics. The indicators remain the same at all instructional levels (K-2, 3-5, 6-8, 9-12), thereby providing an ongoing and constant focus for the standards. The indicators also provide the targets and anchors for broad district-level, program evaluation. State Standard code (e.g. 6.A.1.1.) and standard These statements represent the classroom learning objectives or activities which should be provided at each grade level to help students reach the expectations articulated in the indicators and goals. RCAS curriculum statement with Mastery, Develop, and Introduce symbols (M, D, I)

37 Page 22 6 th Grade Algebra Indicator 1: Use procedures to transform algebraic expressions. 6.A.1.1. Students are able to use order of operations, excluding nested parentheses and exponents, to simplify whole number expressions. (M) use order of operations to evaluate whole number expressions including parenthesis 6.A.1.2. Students are able to write algebraic expressions involving addition or multiplication using whole numbers. (M) write algebraic expressions involving addition and multiplication using whole numbers Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities. 6.A.2.1. Students are able to write and solve one-step 1 st degree equations, with one variable, involving inverse operations using the set of whole numbers. (M) write and solve one-step first degree equations, with one variable, involving inverse operations using the set of whole numbers Indicator 3: Interpret and develop mathematical models. 6.A.3.1. Students are able to identify and graph ordered pairs in Quadrant I on a coordinate plane. (M) identify and graph ordered pairs in Quadrant I on a coordinate plane 6.A.3.2. Students are able to solve one-step problems involving rations and rates. (M) solve one-step problems involving ratios and rates

38 Indicator 4: Describe and use properties and behaviors of relations, functions, and inverses. 6.A.4.1. Students are able to use concrete materials, graphs and algebraic statements to represent problem situations. (M) recognize, describe, and extend arithmetic sequences and patterns (M) use variables to represent given quantities in problem situations The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) understand the significance of the equality and inequality signs (D) identify associative, commutative, distributive, and identity properties involving algebraic expressions (D) identify and graph an inequality on a number line (D) model and solve multi-step problems using unit rates (D) recognize one-step patterns using tables, graphs, and models (I) evaluate powers and exponents (I) solve one-step equations using integers Page 23

39 6 th Grade Geometry Page 24 Indicator 1: Use deductive and inductive reasoning to recognize and apply properties of geometric figures. 6.G.1.1. Students are able to identify and describe the characteristics of triangles and quadrilaterals. (M) identify, classify, and describe characteristics of: o triangles by angle acute, obtuse, right o triangles by side lengths scalene, isosceles, equilateral o quadrilaterals by shape parallelogram, rectangle, rhombus, square 6.G.1.2. Students are able to identify and describe angles. (M) identify, classify, and describe angles by size acute, obtuse, right Indicator 2: Use properties of geometric figures to solve problems from a variety of perspectives. 6.G.2.1. Students are able to use basic shapes to demonstrate geometric concepts. (M) use basic shapes to demonstrate geometric concepts of: o symmetry, reflections, congruency, similarity, perpendicular lines, and parallel lines The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) identify the parts of circles radius and diameter (D) identify, describe, and classify polygons up to ten sides (D) identify relationships among triangles (D) identify relationships among quadrilaterals (D) sketch two-dimensional figures (D) identify and describe elements of geometric figures--altitudes, midpoints, bisectors, radii, diameters, and chords (D) demonstrate ways that shapes can be transformed (flips/reflections, slides/translations, and turns/rotations) (I) describe and classify prisms, pyramids, cylinders, and cones (faces, vertices, edges) (I) use proportions to solve problems (I) apply the properties of triangles and quadrilaterals to find unknown parts

40 6 th Grade Measurement Indicator 1: Apply measurement concepts in practical applications. 6.M.1.1. Select, use, and convert appropriate unit of measurement for a situation. select, use, and convert appropriate units of measurement for a situation. o determine elapsed time o convert length, capacity, and mass within the Metric System (kilo-, base unit, centi-, milli-) o convert weight and length within U.S. Customary System 6.M.1.2. Find the perimeter and area of squares and rectangles (whole number measurements). find the perimeter and area of squares and rectangles (whole number measurements). o apply strategies and/or formulas o use appropriate unit of measure The following are skills that need to be developed or introduced for the subsequent level(s): (D) select, use, and convert units of measurement o measure angles using a protractor o measure and convert length, capacity, and mass, using the U.S. Customary System or the Metric System (kilo- thru milli-) (D) find circumference, perimeter, and area of circles, triangles, and trapezoids, when given the formulas o use appropriate units of measure (whole numbers, decimals, and fractions) o estimate the area of irregular shapes (I) examine measurement situations to determine necessary degree of accuracy (I) find sums and differences of various units within a measurement system (yards, feet, inches) (I) use suitable units when describing rate of change (I) apply proportional reasoning to solve measurement problems with whole number measurements o use scale drawings to represent situations (I) calculate area, volume, and surface area with whole number measurements o apply strategies and/or formulas o find volume of rectangular prisms, rectangular pyramids, cylinders, and cones o calculate area and surface area of composite shapes Page 25

41 Page 26 6 th Grade Number Sense Indicator 1: Analyze the structural characteristics of the real number system and its various subsystems. Analyze the concept of value, magnitude, and relative magnitude of real numbers. 6.N.1.1. Students are able to represent fractions in equivalent forms and convert between fractions, decimals, and percents using halves, fourths, tenths, hundredths. (M) compare and order decimal numbers to the ten-thousandths place (M) represent fractions in equivalent forms and convert between fractions, decimal and percents using halves, fourths, tenths and hundredths (M) convert between mixed numbers and improper fractions o identify both standard and word forms (millions to ten-thousands) of positive rational numbers 6.N.1.2. Students are able to find factors and multiples of whole numbers. (M) find factors (including prime factorization) and multiples of whole numbers (M) classify numbers as prime or composite o using the divisibility rules for 2, 3, 5, 6, 9, and 10 Indicator 2: Apply number operations with real number and other number systems. 6.N.2.1. Students are able to add, subtract, multiply and divide decimals. (M) add, subtract, multiply, divide numbers in decimal form to the ten- thousandths place Indicator 3: Develop conjectures, predictions, or estimations to solve problems and verify or justify the results. 6.N.3.1. Students are able to use various strategies to solve one- and two-step problems involving positive decimals. use various strategies to solve one and two step problems involving positive decimals o rounding off numbers in decimal form to the ten-thousandths place o formulating rules to solve practical problems

42 The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) find the least common denominator of two or more fractions (D) express fractions in simplest form (D) add, subtract, multiply, and divide positive, proper fractions and mixed numbers with like denominators (D) compare and order integers (D) model adding and subtracting integers (D) understand the meaning and effects of operations on rational numbers (I) add, subtract, multiply, and divide positive, proper fractions and mixed numbers with unlike denominators (I) solve consumer application problems involving discounts, mark-ups, commissions, profits, and simple compound interest Page 27

43 6 th Grade Statistics & Probability Page 28 Indicator 1: Use statistical models to gather, analyze, and display data to draw conclusions. 6.S.1.1. Students are able to find the mean, mode, and range of an ordered set of positive data. (M) find the mean, median, mode, and range of an ordered set of positive data 6.S.1.2. Students are able to display data using bar and line graphs, and draw conclusions from data displayed in a graph. (M) read and draw conclusions from tables and graphs (M) solve problems by organizing data into tables and graphs (M) construct bar and line graphs using data Indicator 2: Apply the concepts of probability to predict events/outcomes and solve problems. 6.S.2.1. Students are able to find the probability of a simple event. (M) recognize the probability of a simple event. (i.e. coin flip, single die roll, spinner card) The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) use a variety of visual representations to make predictions. (i.e. bar graphs, line plots, circle graphs, stem-and-leaf plots) (D) express probability as a ratio, decimal, and percent (D) make predictions based on the probability of an independent event o possible outcome from flipping two coins simultaneously (D) calculate the sample space of an independent event (I) use the most appropriate type of central tendency (I) interpret circle graphs and stem-and-leaf plots (I) plot ordered pairs on a scatter plot (I) organize data into frequency tables (I) compute the probability for two simultaneous, independent events

44 7 th Grade Algebra Indicator 1: Use procedures to transform algebraic expressions. 7.A.1.1. Students are able to write and evaluate algebraic expressions using the set of whole numbers. (M) use order of operations to evaluate whole number algebraic expressions using exponents and parenthesis 7.A.1.2. Students are able to identify associative, commutative, distributive, and identity properties involving algebraic expressions. (M) identify and use the associative, commutative, distributive, and identity properties involving algebraic expressions Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities. 7.A.2.1. Students are able to write and solve one-step 1 st degree equations, with one variable, using the set of integers and inequalities, with one variable, using the set of whole numbers. (M) write simple expressions using variables (convert from word phrases to symbols) (M) write and solve one-step equations and inequalities using integers Indicator 3: Interpret and develop mathematical models. 7.A.3.1. Students are able to identify and graph ordered pairs on a coordinate plane and inequalities on a number line. (M) identify and graph ordered pairs on a coordinate plane (M) solve and graph simple one-step inequalities 7.A.3.2. Students are able to model and solve multi-step problems involving rates. (M) model and solve multi-step problems involving rates (best buy or fastest speed or greatest rate) Page 29

45 Page 30 Indicator 4: Describe and use properties and behaviors of relations, functions, and inverses. 7.A.4.1. Students are able to recognize one-step patterns using tables, graphs, and models, and create one-step algebraic expressions representing the pattern. (M) write algebraic statements to represent patterns observed in real life situations using tables, graphs, and models The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) use the associative, commutative, distributive, and identity properties to expand and simplify algebraic expressions with the set of integers (D) identify x- and y-intercepts from a graph (I) use order of operations with nested parenthesis (I) write and solve two-step equations (I) identify positive and negative slope (i.e. steepness)

46 7 th Grade Geometry Indicator 1: Use deductive and inductive reasoning to recognize and apply properties of geometric figures. 7.G.1.1. Students are able to identify, describe, and classify polygons having up to ten sides. (M) identify, describe, and classify polygons up to ten sides o relationships among triangles o relationships among quadrilaterals o sketch two-dimensional figures 7.G.1.2. Students are able to identify and describe elements of geometric figures. (M) identify and describe elements of geometric figures altitudes, midpoints, bisectors, radii, diameters, and chords Indicator 2: Use properties of geometric figures to solve problems from a variety of perspectives. 7.G.2.1. Students are able to demonstrate ways that shapes can be transformed. (M) demonstrate ways that shapes can be transformed (flips/reflections, slides/translations, and turns/rotations) The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) describe and classify prisms, pyramids, cylinders, and cones (faces, vertices, edges) (D) use the Pythagorean Theorem to find the missing side of a right triangle (with the set of whole numbers) (D) write and solve proportions that express relationships between corresponding parts of similar quadrilaterals and triangles (D) manipulate tessellations (D) apply the properties of triangles and quadrilaterals to find unknown parts (I) identify three-dimensional shapes using nets (I) recognize and identify the relationships between angles created by intersecting lines using the concepts of complementary, supplementary, vertical, and corresponding angles (I) identify corresponding parts of congruent and similar triangles and polygons (I) use proportions to solve problems Page 31

47 7 th Grade Measurement Page 32 Indicator 1: Apply measurement concepts in practical applications. 7.M.1.1. Select, use, and convert appropriate unit of measurement for a situation including capacity and angle measurement. select, use, and convert appropriate units of measurement for a situation including capacity and angle measurement o measure angles o measure length, capacity, and mass o convert within the Metric System (kilo- thru milli-) o convert within the U.S. Customary System (weight, length, capacity) 7.M.1.2. Given the formulas, find the circumference, perimeter, and area of circles, parallelograms, triangles, and trapezoids (whole number measurements). find circumference, perimeter, and area of circles, triangles, and trapezoids, when given the formulas (whole number measurements) o use appropriate units of measure o estimate the area of irregular shapes The following are skills that need to be developed or introduced for the subsequent level(s): (D) apply proportional reasoning to solve measurement problems with rational number measurements o convert within measurement systems o use scale drawings o calculate indirect measurement (D) find area, volume, and surface area with whole number measurements o use appropriate units of measure o apply strategies and/or formulas o calculate area of composite shapes (D) use suitable units when describing rate of change (D) examine measurement situations to determine necessary degree of accuracy (I) use indirect measurement in problem situations that defy direct measurement (I) choose appropriate unit label, scale, and precision o determine appropriate scales for histograms, scatter plots, and other graphs

48 7 th Grade Number Sense Indicator 1: Analyze the structural characteristics of the real number system and its various subsystems. Analyze the concept of value, magnitude, and relative magnitude of real numbers. 7.N.1.1. Students are able to represent numbers in a variety of forms by describing, ordering, and comparing integers, decimals, percents, and fractions. (M) represent numbers in a variety of forms by describing, ordering, and comparing integers, decimals, fractions, and percents o identifying and representing terminating and repeating decimals (M) determine whether ratios are equal and proportional 7.N.1.2. Students are able to find and use common multiples and factors of whole numbers. (M) find and use common factors and multiples of whole numbers o calculate the GCF and LCM using prime factorization o use the divisibility rules for 2, 3, 4, 5, 6, 9, and 10 Indicator 2: Apply number operations with real numbers and other number systems. 7.N.2.1. Students are able to add, subtract, multiply, and divide integers and positive fractions. (M) add subtract, multiply, and divide integers and positive fractions (with like and unlike denominators) Indicator 3: Develop conjectures, predictions, or estimations to solve problems and verify or justify the results. 7.N.3.1. Students are able to use various strategies to solve one- and two-step problems involving positive fractions and integers. (M) use various strategies to solve one and two step problems involving positive fractions and integers Page 33

49 Page 34 The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) find the absolute value of integers (D) represent numbers in a variety of equivalent forms (positive and negative exponents, scientific notation) (D) develop the meaning of percents less than one and greater than one hundred (D) identify, represent, compare, and order rational numbers and represent them on a number line (D) solve consumer application problems involving discounts, mark-ups, commissions, profits, and simple compound interest (D) use estimation strategies to make predictions and test the reasonableness of the answer (D) formulate rules to solve practical problems involving rational numbers (D) use proportions to solve a variety of problems (I) find the absolute value of integers (I) use mathematical symbols to represent irrational numbers (i.e. square root symbol,, and bar notation) (I) formulate counter examples to disclaim given assertions (I) compare and order radicals, numbers expressed with exponents, and the absolute values of various numbers

50 7 th Grade Statistics & Probability Page 35 Indicator 1: Use statistical models to gather, analyze, and display data to draw conclusions. 7.S.1.1. Students are able to find the mean, median, mode and range of a set of data. (M) find the mean, median, mode, and range of an unordered set of data 7.S.1.2. Students are able to display data using frequency tables, line plots, stem-and-leaf plots, and make predictions from data displayed in a graph. (M) interpret and construct bar graphs, line plots, circle graphs, and stem and leaf plots using appropriate scales, labels, and intervals (M) solve and analyze problems by organizing data into tables and frequency charts (M) use a variety of visual representations to make predictions. (i.e. bar graphs, line plots, circle graphs, stem and leaf plots) Indicator 2: Apply the concepts of probability to predict events/outcomes and solve problems. 7.S.2.1. Students are able, given a sample space, to find the probability of a specific outcome. (M) calculate probability, given a sample space, with a known number of outcomes o possible outcome from flipping two coins simultaneously The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) plot ordered pairs on a scatter plot (D) compute the probability of two simultaneous, independent events (I) use central tendency in real world situations (I) identify the line of best fit (line of regression) on a scatter plot to determine the type of correlation (I) recognize the difference between experimental and theoretical probabilities (I) use various ways to calculate the number of possible outcomes (The Fundamental Counting Principle) (I) understand the concept of sampling bias and ways to select unbiased samples

51 Page 36 Indicator 1: Use procedures to transform algebraic expressions. 8 th Grade Algebra 8.A.1.1. Students are able to use properties to expand, combine, and simplify 1 st degree algebraic expressions with the set of integers. (M) use the commutative, associative, distributive, and identity properties to expand and simplify algebraic expressions with the set of integers (M) use order of operations with exponents and nested parentheses to simplify algebraic expressions with integers (M) determine if two 1 st degree algebraic expressions are equivalent Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities. 8.A.2.1. Students are able to write and solve two-step 1 st degree equations, with one variable, and one-step inequalities, with one variable, using the set of integers. (M) write and solve two-step equations using the set of integers (M) write and solve one-step inequalities using the set of integers Indicator 3: Interpret and develop mathematical models. 8.A.3.1. Students are able to describe and determine linear relationships. (M) describe and determine linear relationships (determine slope from a line or ordered pairs on a graph, identify x- and y-intercepts from a graph)

52 Indicator 4: Describe and use properties and behaviors of relations, functions, and inverses. 8.A.4.1. Students are able to create rules to explain the relationship between numbers when a change in the first variable affects the second variable. (M) explain how a change in one variable affects the other variable (when given x + 10 = y or y = 6x, as x increases what happens to y) 8.A.4.2. Students are able to describe and represent relations using tables, graphs, and rules. (M) describe and represent relations using tables, graphs, and rules (using function tables, number machines, or input-output tables) The following are skills that need to be developed or introduced for the subsequent level(s): (D) write equivalent forms of algebraic expressions using properties of the set of real numbers (D) create linear models to represent problem situations (I) be able to distinguish between linear and nonlinear models (I) be able to use graphs, tables, and equations to represent linear functions (create a table from a graph or an equation of a line) Page 37

53 8 th Grade Geometry Page 38 Indicator 1: Use deductive and inductive reasoning to recognize and apply properties of geometric figures. 8.G.1.1. Students are able to describe and classify prisms, pyramids, cylinders, and cones. (M) describe and classify prisms, pyramids, cylinders, and cones (faces, vertices, edges) (M) construct three-dimensional figures from two-dimensional views using nets 8.G.1.2. Students, when given any two sides of an illustrated right triangle, are able to use the Pythagorean Theorem to find the third side. (M) calculate the third side of a right triangle when given two sides, using the Pythagorean Theorem (with the set of whole numbers) Indicator 2: Use properties of geometric figures to solve problems from a variety of perspectives. 8.G.2.1. Students are able to write and solve proportions that express the relationships between corresponding parts of similar quadrilaterals and triangles. (M) write and solve proportions that express relationships between corresponding parts of similar quadrilaterals and triangles The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) use visual perspective to analyze geometric problems (construct three-dimensional models given top, side, or bottom views of objects) (D) create two-dimensional representations that demonstrate various perspectives of three-dimensional objects (D) recognize and identify the relationships between angles created by intersecting lines using the concepts of complementary, supplementary, vertical, and corresponding angles (D) identify corresponding parts of congruent and similar triangles and polygons (D) apply the properties of triangles and quadrilaterals to find unknown parts (I) use parallel lines cut by a transversal to identify the congruent angles created

54 8 th Grade Measurement Indicator 1: Apply measurement concepts in practical applications. 8.M.1.1. Apply proportional reasoning to solve measurement problems with rational number measurements. apply proportional reasoning to solve measurement problems with rational number measurements o convert within measurement systems o use scale drawings o calculate indirect measurement 8.M.1.2. Find area, volume, and surface area with whole number measurements. find area, volume, and surface area with rational number measurements o use appropriate units of measure o apply strategies and/or formulas o calculate volume of rectangular prisms, rectangular pyramids, cylinders, and cones o calculate area and surface area of composite shapes The following are skills that need to be developed or introduced for the subsequent level(s): (D) choose appropriate unit label, scale, and precision o determine appropriate scales for histograms, scatter plots, and other graphs (D) use suitable units when describing rate of change (D) use formulas to find perimeter, circumference, and area to solve problems involving common geometric figures o use algebraic expressions with geometric formulas (D) use indirect measurement in problem situations that defy direct measurement (I) use dimensional analysis to check answers and determine units of a problem solution o convert rate of change between measurement systems o 2 cups * 2 pints * 4 quarts = 16 cups 1 pint 1 quart 1 gallon 1 gallon Page 39

55 8 th Grade Number Sense Indicator 1: Analyze the structural characteristics of the real number system and its various subsystems. Analyze the concept of value, magnitude, and relative magnitude of real numbers. 8.N.1.1. Students are able to represent numbers in a variety of forms, and identify the subsets of rational numbers. (M) represent numbers in a variety of forms (radicals, absolute value, exponents, and scientific notation) (M) identify the subsets of rational numbers (counting/natural, whole, integers) Indicator 2: Apply number operations with real numbers and other number systems. 8.N.2.1. Students are able to read, write, and compute within any subset of rational numbers. (M) read, write and compute within any subset of rational numbers (M) solve consumer application problems involving discounts, mark-ups, commissions, profits, and simple interest Indicator 3: Develop conjectures, predictions, or estimations to solve problems and verify or justify the results. 8.N.3.1. Students are able to use various strategies to solve multi-step problems involving rational numbers. (M) use various strategies to solve multi-step problems involving rational numbers (M) formulate rules to solve practical problems involving rational numbers (M) use estimation strategies to make predictions and test the reasonableness of the answer (M) simplify expressions using properties to justify steps The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) solve inequalities that require multiplying and dividing by a negative number (D) use multiple representations of a given real number (D) compare and order radicals, numbers expressed with exponents, and the absolute values of various numbers (I) identify and classify irrational numbers in the real number system and find their approximate location on the number line (I) identify the various subsets of the real number system (rational and irrational numbers) Page 40

56 Page 41 8 th Grade Statistics & Probability Indicator 1: Use statistical models to gather, analyze, and display data to draw conclusions. 8.S.1.1. Students are able to find mean, median, mode, and range of a data set from a stem-and-leaf plot and a line plot. (M) systematically collect, organize, describe, and analyze sets of data to make simple predictions using mean, median, mode, and range on stem-and-leaf plots and line plots 8.S.1.2. Students are able to use a variety of visual representations to display data to make comparisons and predictions. (M) use a variety of visual representations to display data to make comparisons and predictions (i.e. double-bar graphs, double-line graphs, and scatter plots) Indicator 2: Apply the concepts of probability to predict events/outcomes, and solve problems. 8.S.2.1. Students are able to find the sample space and compute probability for two simultaneous, independent events. (M) find the sample space and compute probability of two simultaneous, independent events The following are skills that need to be developed or introduced for the subsequent grade level(s): (D) identify the line of best fit (line of regression) on a scatter plot to determine the type of correlation (positive, negative, or no correlation) (D) understand the difference between experimental and theoretical probabilities (D) use various ways to calculate the number of possible outcomes (The Fundamental Counting Principle) (D) understand the concept of sampling bias and ways to select unbiased samples (I) explore the interquartile range using box-and-whisker plots (I) calculate the probability of a compound event

57 RCAS Algebra I Curriculum

58 Page 42 RCAS Curriculum Algebra I *Bold italicized type indicates Core SD Standards bold type indicates Advanced SD State Standards. I. PROPERTIES OF REAL NUMBERS AND EXPRESSIONS The learner will master: A. (9-12.A.1.1) Write equivalent forms of algebraic expressions using properties of the set of real numbers. B. (9-12.G.2.3) Use proportions to solve problems. C. (9-12.N.1.1) Identify multiple representation of a real number. D. (9-12.N.1.2) Apply the concept of place value, magnitude, and relative magnitude of real numbers. E. (9-12.N.2.1) Add, subtract, multiply, and divide real numbers including integral exponents. The learner will be introduced to: F. (9-12.A.1.1A) Write equivalent forms of rational algebraic expressions using properties of real numbers. G. (9-12.N.1.2A) Apply properties and axioms of the real number system to various subsets, e.g., axioms of order, closure. H. (9-12.G.1.4A) Use formulas for surface area and volume to solve problems involving threedimensional figures.

59 II. LINEAR EQUATIONS AND INEQUALITIES The learner will master: A. (9-12.A.2.1) Use algebraic properties to transform multi-step, single variable, and first-degree equations. B. (9-12.A.2.2) Use algebraic properties to transform multi-step, single variable, and first-degree inequalities and represent solutions using a number line. The learner will develop: C. (9-12.A.2.3A) Determine solutions to absolute value statements. D. (9-12.A.4.6A) Graph solutions to linear inequalities. The learner will be introduced to: E. (9-12.A.2.2A) Determine the solution of systems of equations and systems of inequalities. III. LINEAR MODELING The learner will master: A. (9-12.A.3.1) Create linear models to represent problem situations. B. (9-12.A.3.2) Distinguish between linear and nonlinear models. C. (9-12.A.3.1A) Distinguish between linear, quadratic, inverse variations, and exponential models. D. (9-12.N.3.1) Use estimation strategies in problem situations to predict results and to check the reasonableness of results. E. (9-12.N.3.2) Select alternative computational strategies and explain the chosen strategy. Page 43

60 Page 44 F. (9-12.M.1.1) Choose appropriate unit label, scale, and precision. G. (9-12.M.1.2) Use suitable units when describing rate of change. H. (9-12.M.1.3) Use formulas to find perimeter, circumference, and area to solve problems involving common geometric figures. The learner will develop: I. (9-12.A.3.2A) Create formulas to model relationships that are algebraic, geometric, trigonometric, and exponential. J. (9-12.M.1.1A) Use dimensional analysis to check answers and determine units of a problem solution. The learner will be introduced to: K. (9-12.A.3.3A) Use sequences and series to model relationships. IV. FUNCTIONS AND PATTERNS The learner will master: a. (9-12.A.4.1) Use graphs, tables, and equations to represent linear functions. The learner will develop: b. (9-12.A.4.1A) Determine the domain, range, and intercepts of a function. The learner will be introduced to: c. (9-12.A.4.3A) Apply transformations to graphs and describe the results.

61 V. QUADRATICS The learner will be introduced to: A. Add, subtract, multiply, divide, and simplify radicals. B. Factor polynomials using common factors, difference of two perfect squares, grouping, perfect square trinomials, and trial and error. C. Introduce the quadratic formula. D. (9-12.A.2.1A) Determine solutions of quadratic equations. VI. STATISTICS AND PROBABILITY The learner will master: A. (9-12.S.1.1) Draw conclusions from a set of data. B. (9-12.S.1.2) Compare multiple one-variable data sets, using range, interquartile range, mean, mode, and median. C. (9-12.S.1.3) Represent a set of data in a variety of graphical forms and draw conclusions. D. (9-12.S.2.1) Distinguish between experimental and theoretical probability. E. (9-12.S.2.2) Predict outcomes of simple events using given theoretical probabilities. The learner will be introduced to: F. (9-12.S.1.2A) Analyze and evaluate graphical displays of data. G. (9-12.S.1.5A) Use scatterplots, best-fit lines, and correlation coefficients to model data and support conclusions. H. (9-12.S.2.1A) Use probabilities to solve problems. I. (9-12.S.2.3A) Generate data and use the data to estimate empirical probabilities. Page 45

62 Core High School Algebra Performance Descriptors Page 46 Advanced Proficient Basic High school students performing at the advanced level: transform algebraic expressions solve quadratic equations solve a system of linear equations High school students performing at the proficient level: transform polynomial expressions using real number properties solve single variable linear equations with integral coefficients graph linear equations interpret tables, graphs, and charts to solve problems create a linear model from a problem context High school students performing at the basic level: transform linear expressions with integral coefficients using real number properties solve linear equations of the form recognize the graph of a linear equation graph a line from a table of values ax + b = c, where a, b, and c are integers

63 SD Abbreviated Math Standards with examples

64 SD Standards Glossary

65 Page 56 SOUTH DAKOTA MATHEMATICS STANDARDS GLOSSARY *Note: This glossary contains explanations, not necessarily formal mathematical definitions, of terms used in the standards document. Absolute value: A number s distance from 0 on the number line. The absolute value of -4 is 4; the absolute value of 4 is 4; the symbol is n. Acute angle: An angle whose measure is more than 0 but less than 90. Algorithm: An organized sequential procedure for performing a given type of calculation or solving a given type of problem. An example is long division. Analog: Having to do with data represented by continuous variables, e.g., a clock with hour, minute, and second hands. Area: The measure of a region of a plane, usually represented by the number of square units needed to cover a surface enclosed by a geometric figure. Arithmetic sequence: A sequence of elements in which each term is the result of adding a fixed number to the previous term. Associative property: Allows numbers to be regrouped in an addition or multiplication problem, e.g., a x (b x c) = (a x b) x c. a + ( b + c) = ( a + b) + c ; Axiom: A basic assumption about a mathematical system from which theorems can be deduced. For example, the system could be the points and lines in the plane. Then an axiom would be that given any two distinct points in the plane, there is a unique line through them. Bar graph: A graph form using rectangular bars to summarize data; specifically to show how many observations fall into a particular category.

66 Box-and-whisker plot: A graphical method for displaying the median, quartiles, and extremes of a set of data, using the number line. Cartesian plane: See coordinate plane. Circle: A set of all points in a plane that are the same distance from a given point in the plane. Circumference: Distance around a circle. The formula is: C = 2 r, where r is the circle s radius, or C = d, where d is the circle s diameter. Closure property: A set of numbers, such as the integers, is closed under a particular operation if performing the operation on numbers in the set results in another number in that set. For example, the set of non-zero integers is closed under multiplication, but is not closed under division. Coefficient: The numerical part of a term, e.g., 5 is the coefficient of the 2 x term in x. Combination: A selection in which order is not considered. Commutative property: The property of a number system that provides for the reordering of terms in certain operations, such as addition and multiplication, e.g., a + b = b + a, ab = ba. Compensation: A mental math strategy in which one addend is changed to a multiple of 10 and then the other addend is adjusted to keep the balance, e.g., (16 1) + (9 + 1) (7 1) + (9 + 1) = = 16 Complex number: A number of the form a + bi where a and b are real numbers and i = 1. Cone: A three-dimensional shape in space that has a circular base and one vertex. Page 57

67 Page 58 Congruent: Geometric figures or angles that have the same size and shape. Two angles are congruent if they have the same measure. Two line segments are congruent in they have the same length. The symbol is. Conjecture: An informed, educated guess. Composite number: A natural number greater than one that is not prime. Conversion factor: A numerical factor used to multiply or divide a quantity when converting from one system of units to another. Coordinate plane: A plane in which two number lines called coordinate axes intersect at right angles and are usually called the x-axis and y-axis. Every point in a coordinate plane can be described uniquely by an ordered pair of numbers, the coordinates of the point with respect to the coordinate axes. Cosine: The cosine of an angle ( ), cos( ) is the x-coordinate of the point on the unit circle so that the ray connecting the point with the origin makes an angle of with the positive x-axis. When q is an angle of a right triangle, then cos( ) is the ratio of the adjacent side to the hypotenuse. Counting number: A number used for counting objects, i.e., a number from the set {1, 2, 3, 4, 5, }. The same as natural numbers. Decimal number: A numeral that contains a decimal point, such as Deductive reasoning: A type of reasoning in which the conclusion about particulars follows necessarily from general or universal premises. Difference: The result of a subtraction. Digit: Any of the symbols used to write numbers, 0-9. The ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The number 215 has three digits: 2, 1, and 5. Digital: Having to do with data that is represented in the form of numerical digits; providing a readout in numerical digits, e.g., a digital watch. Dilation: A type of transformation that is a proportional shrinking or enlargement of a figure.

68 Dimensional analysis: A method of manipulating unit measures algebraically to determine the proper units for a quantity computed algebraically. For example, velocity has units of the form distance over time (e.g., meters per second [m/sec]), and acceleration has 2 units of velocity over time; so it follows that acceleration has units ( m /sec)/sec = m /(sec ). Distributive property: A property of the number system in which multiplication distributes over addition represented by a ( b + c) = ab + ac. Domain: The domain of a function is the set of possible values for x. Elapsed time: Difference between starting time and the ending time of an event. Expanded form: The expanded form of an algebraic expression is the equivalent expression without parentheses. For example, the expanded form of ( a + b) is a + 2ab + b. Expanded notation: A way of representing a number that shows the sum of each digit multiplied by the appropriate positive power of 3 2 ten and the units digit, e.g., 3451 as or as Exponent: The number that indicates how many times the base is used as a factor, e.g., in 4 3 = = 64, the exponent is 3, indicating that 4 is repeated as a factor three times. Exponential function: A function commonly used to study growth and decay. It has the form than a = 1. Factors: Any of two or more quantities that are multiplied together, for example, 2 and 3 are factors of 6. First degree expressions: See linear expressions. First degree equations: An equation involving only first degree expressions. x y = a with a positive value of a other Five number summary: For a data set, the numbers representing the minimum and maximum values, first and third quartiles, and median. Page 59

69 Page 60 Function: A mathematical relation that associates each object in a set with exactly one value. Fundamental counting principle: If event M can occur in m ways, and after it has occurred, event N can occur in n ways, then event M followed by event N can occur m n ways. Geometric pattern: A sequence of symbols or geometric figures. Geometric sequence (progression): An ordered set of numbers that has a common ratio between consecutive terms, e.g., {1, 3, 9, 27, 81 }. Histogram: A vertical block graph with no spaces between the blocks. It is used to summarize data by representing the frequency of observations that fall within uniform intervals of values. Hypotenuse: In a right triangle the hypotenuse is the longest side which is opposite the right angle. Identity property: Adding 0 to a number does not change the value. Multiplying a number by 1 does not change the value. Independent event: Two events in which the occurrence of one event does not affect the probability of the occurrence of the other. Inductive reasoning: The type of reasoning that uses inference to reach a generalized conclusion from particular instances. Inequality: A relationship between two quantities involving one of the following relationships: less than (<), less than or equal to ( ), greater than (>), greater than or equal to ( ), or not equal ( ). Integer: A number that is either a whole number or the negative of a whole number. Integral: Refers to an integer. Intercepts: The values where the graph of a relation crosses the axes. Interquartile range: The difference between the third and first quartiles. Inverse of a function: f (x) is a function, g (x) such that f ( g( x)) = x and g ( f ( x)) = x.

70 Inverse operations: Subtraction is the inverse operation for addition. Division is the inverse operation for multiplication. Irrational number: A number that cannot be expressed as a quotient of two integers, e.g., cannot be written as a repeating or terminating decimal. 2. A number is irrational if and only if it Landmark numbers: Numbers that are familiar landing places that make for simple calculations and to which other numbers can be related such as 10, 100, and 1,000 and their multiples and factors. Line graph: A graph that connects data points. Line of best fit: A line drawn through, or near to, as many data points as possible on a scatter plot. Line plot: A number line with dots or other marks above it to show the number of times an event occurs. Linear equation: Any equation that can be written in the form Ax + By + C = 0 where A and B cannot both be 0. The graph of such an equation is a line. Linear function: A function of the form f ( x) = mx + b where m and b are some fixed numbers, representing slope and y-intercept. Functions of this kind are called linear because their graphs are lines. Linear expression: An expression of the form constants. ax + by + c, ax + by + cz + d, where x, y, and z variable and a, b, c, and d are Linear pattern: See arithmetic sequence. Linear relationship: A relationship involving only linear expressions. Line of symmetry: A line that divides a figure into two halves that are mirror images of each other. Logarithm: Another way to express an exponent. For example, if 10 2 = 100 then log = 2. Mean: In statistics, the average obtained by dividing the sum of two or more quantities by the number of these values. Page 61

71 Page 62 Measures of central tendency: The mean, median, and mode of a set of data. Median: In statistics, the quantity designating the central value in a set of numbers. The center number (or the average of the two central numbers) of a list of data when the numbers are arranged in order from the least to greatest. Mode: In statistics, the value that occurs most frequently in a given set of numbers. Monomial: A product of numbers and/or variables, e.g., Natural numbers: The set of counting numbers. 2 5x, 3 x 2 y, 7 yz 3 2 x. Nested parentheses: Grouping symbols within grouping symbols, [ 10(3 + 2) 12]. Net: Two-dimensional pattern that can be folded to form a three-dimensional shape. Nonstandard unit: Unit of measurement expressed in terms of objects (such as paper clips, sticks of gum, shoes, etc.). Numeral: A symbol, not a variable, that is used to represent a number. Also known as a number. Numerical expressions: An expression using only numerals (numbers). Numeric pattern: A pattern composed of numerals (numbers). Obtuse angle: An angle with a measurement greater than 90 and/or less than 180. Operational symbols: Symbols used to indicate operations, such as + for addition, x for multiplication, etc. Order of operations: Rules that describe the sequence used in evaluating expressions. In this order: parenthesis, exponents, multiplication and division, addition and subtraction. Ordered pair: A pair of numbers that gives the coordinates of a point on a coordinate plane in this order (horizontal coordinate, vertical coordinate).

72 Ordinal number: A number designating the place (as first, second, or third) occupied by an item in an ordered sequence. Parallel: Given, distinct lines in the plane that are infinite in both directions, the lines are parallel if they never meet. Perimeter: The distance around a plane, closed geometric figure. Permutation: A permutation is a specific reordering of a set of numbers {1, 2,..., n}. Perpendicular lines: Two lines that intersect at right angles. Pictogram (pictograph): A graph that uses pictures to show and compare information. Plane: A flat surface that extends indefinitely in all directions. Polynomial: In algebra, a sum of monomials; for example, xy y. x + Prime Number: A natural number greater than 1 is prime if and only if the only positive integer factors are 1 and the number itself. The first seven primes are 2, 3, 5, 7, 11, 13 and 17. Probability: A number from 0 to 1 that describes the likelihood that a given event will take place. For example, the probability of throwing a 6 with a single throw of one die is 1/6. Product: The result of multiplication. Page 63

73 Page 64 Proof: A method of constructing a valid argument, using deductive reasoning. Proportion: An equation that states that two ratios are equivalent, e.g., 4 1 = or 4 : 8 = 1: Pythagorean Theorem: For any right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse, e.g., a + b = c where a and b are the legs and c is the hypotenuse. Quadrant: One of the four regions into which the coordinate plane is divided. Quadratic function: A function containing 2 x, a polynomial of degree 2 such as 2 f ( x) = ax + bx + c. Quartile: The value of the boundary at the 25 th, 50 th, or 75 th percentiles of a frequency distribution divided into four parts, each containing a quarter of the population. Quotient: The result of a division. Random sample: A group of people or objects chosen from a larger group or population by a process giving equal chance of selection to all possible people or objects. Range: In statistics, the difference between the greatest and smallest values in a data set. Range of a function: The set of possible values for y or f (x). Rate: A ratio that compares two quantities measured in different units. Ratio: A quotient of two numbers or like quantities, e.g., 4 to 7 or 4 : 7 or Rational number: A number that can be written as the ratio of two integers, e.g., 0.5,, -3, 8, Real number: The set of numbers consisting of all rational and all irrational numbers.

74 Reflection: A type of transformation that creates a mirror image of a figure on the opposite side of a line, called the line of symmetry. Relation: An equation that expresses the relationship between two variables. Right angle: An angle with a measurement of 90. Root: A number that can be used as a factor a given number of times to produce the original number; for example, the fifth root of 32 = 2 because =. Root of an equation: A value that makes the equation true. Rotation: A type of transformation that turns a figure about a fixed point, called the center of rotation. Sample space: In probability, the set of all possible outcomes of a given experiment, e.g., the sample space for tossing two coins is {(H,H), (H,T), (T,H), (T,T)}. Scalene triangle: A triangle with three unequal sides. Scatter plot: Two sets of data plotted as ordered pairs in the coordinate plane. Winning Olympic High Jump Scientific notation: A system in which numbers are expressed as products consisting of a number from 1 to 10 multiplied by an appropriate power of 10, e.g., 562 = 5.62 x Page 65

75 Page 66 Sequence: A set of elements that can be counted, e.g., 1, 3, 9, 27, 81. In this sequence, 1 is the first term, 3 is the second term, 9 is the third term, and so on. Similarity: Having the same shape but not necessarily the same size. Sine: The sine of an angle θ (sin θ ) is the y-coordinate of the point on the unit circle so that the ray connecting the origin to the point makes an angle of θ with the positive x-axis. When θ is an angle of a right triangle, then sin(θ ) is the ratio of the opposite side with the hypotenuse. Single variable equation: An equation with one variable. Square number: The product when a whole number is multiplied by itself. Square root: A number n is a root of a number m if 2 n = m. The square root of 16 is 4 or 4. Standard deviation: A statistic that measures the dispersion of a sample. Sum: The result of addition. Symmetry: A figure has symmetry when one side is the mirror image of the other side. System of linear equations: Two or more linear equations used to describe a situation. Tangent: A line, curve, or surface meeting another line, curve, or surface at a single point and sharing a common tangent line or tangent plane at that point. The tangent of an angleθ, tanθ, is the ratio of sin θ to cosθ. In a right triangle, tan θ is the ratio of the opposite side length to the adjacent side length. Tessellation: A repetitive pattern of polygons that fit together with no gaps or overlaps. Transformation: A rule that sets up a one-to-one correspondence between the points in a geometric object (the pre-image) and the

76 points in another geometric object (the image). Reflections, rotations, translations, and dilations are particular examples of transformations. Translation: Sliding a figure from one position to another without turning or flipping the figure. Transversal: In geometry, a line (k) that intersects two or more lines (m and n) at different points. k m n Unit fraction: A fraction whose numerator is 1 (e.g.,,, ) Unit rate: A rate with a denominator of 1. Variable: A letter or symbol used to represent one or more numbers in an expression, equation, inequality, or matrix. Venn diagram: A diagram that is used to show relationships between sets. Page 67