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KDS COURSE #6: “Discovery-Based Mathematics” Course Provider: Knowledge Delivery Systems Website: www.kdsi.org Address: 110 William Street, Floor 32 New York, NY 10038-3091 Contact Person(s): Contact 1: KDS Email: schools@kdsi.org Phone: (212) 809-2969 Course Title: Discovery-Based Mathematics Course Length: 30 hours (lecture time, plus additional assessment and forum interaction) NOTE: Parts I & II of the course each include 15 hours of streaming video Term: YEAR-ROUND (Fall, Spring, Summer) Time: Self-paced (Weekdays, Weekends, Available 24-7) Content Area(s Mathematics, Sciences (any content area with computational mastery necessary) Target Grade Level(s): Elementary and Early Middle School (Focus on Grade 3-6, teachings can be adapted to relate to other grade levels and subject areas) Description of Course (including references to research, alignment to student achievement, and best practices): This course reflects the NCTM math standards and is delivered through discovery-based strategies and materials which were developed over a two year period for a large urban district which resulted in an 18 point rise in students passing the fourth grade state-wide assessment in mathematics. The course will provide teachers with easy-to-implement, well-sequenced activities that promote conceptual understanding and that relate concrete understanding to symbolic interpretation. Teachers will acquire techniques to assess all students’ understanding of skills and concepts so that lessons can be adjusted to meet student needs and expand their understanding. Teachers will be provided with activities that require pattern recognition and descriptions with respect to operational procedures including whole numbers, fractions and decimals. Teachers will be able to utilize activities that provide creative practice with operational skills. They will be introduced to strategies that promote number sense, estimation strategies, and foundational understanding so that students can chose appropriate and efficient strategies to determine solutions. Note: Requires Discovery-Based Math Manipulatives Kit The supplemental kit includes a custom-tailored workbook to follow the online courses, as well as correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,” along with many hands-on math manipulatives. (The materials are packaged in a convenient carrying case.) Instructional Goals: Participants will know: 1. Reasons why constructivist, alternative strategies are essential elements to be implemented into all math classrooms 2. How to use base-ten blocks, connecting cubes, geoboards, fraction tiles, pattern blocks and rulers to teach conceptual understanding, fact mastery and algorithmic understanding of all four arithmetic operations with whole numbers, fractions and decimals 3. Why all students need to experience concepts through concrete, iconic and then symbolic stages of learning 4. Techniques of how to transfer concrete understanding to fact and algorithmic mastery of traditional algorithms 5. How to use response devices to monitor understanding, adjust lessons and formulate questions that reflect students current level of understanding 6. How to incorporate techniques that build foundational understanding, number sense and estimation skills so that students can effectively know when to use mental math, paper and pencil, or estimation and then a calculator to solve basic computational problems involving whole numbers fractions and decimals. 7. How to use inquiry based techniques to foster student understanding, self esteem and self- confidence. Standards Addressed: The comprehensive course will help teachers effectively develop strategies for balanced mathematics , through hands-on, discovery-based learning so that their students will: Understand the concepts of and become proficient with the skills of mathematics; Communicate and reason mathematically; Become problem solvers by using appropriate tools and strategies; through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability. More specifically: Number Sense and Operations Strand: The course will effectively develop strategies for teachers so that their students will: Understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems; Understand meanings of operations and procedures, and how they relate to one another; 2 Compute accurately and make reasonable estimates. Algebra Strand: The course will effectively develop strategies for teachers so that their students will: Represent and analyze algebraically a wide variety of problem solving situations; Perform algebraic procedures accurately; Recognize, use, and represent algebraically patterns, relations, and functions. Geometry Strand: The course will effectively develop strategies for teachers so that their students will: Use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes; Identify and justify geometric relationships, formally and informally; Apply transformations and symmetry to analyze problem solving situations; Apply coordinate geometry to analyze problem solving situations. Measurement strand: The course will effectively develop strategies for teachers so that their students will: Determine what can be measured and how, using appropriate methods and formulas; Use units to give meaning to measurements; Understand that all measurement contains error and be able to determine its significance; Develop strategies for estimating measurements. Statistics and probability strand: The course will effectively develop strategies for teachers so that their students will: Collect, organize, display, and analyze data; Make predictions that are based upon data analysis; Understand and apply concepts of probability. Problem solving strand: The course will effectively develop strategies for teachers so that their students will: 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 strand: The course will effectively develop strategies for teachers so that their students will: 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. 3 Communication strand: The course will effectively develop strategies for teachers so that their students will: 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 strand: The course will effectively develop strategies for teachers so that their students will: Recognize and use connections among mathematical ideas; Understand how mathematical ideas interconnect and build on one another to produce a coherent whole; Recognize and apply mathematics in contexts outside of mathematics. Representation strand: The course will effectively develop strategies for teachers so that their students will: 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. (Within the Content Strands) Number Sense and Operations Number Systems Number Theory Operations Estimation Algebra Variables and Expressions Equations and Inequalities Patterns, Relations, and Functions Coordinate Geometry Trigonometric Functions (for lower level trigonometry) Geometry Shapes Geometric Relationships Transformational Geometry Coordinate Geometry Constructions Measurement Units of Measurement Tools and Methods Units 4 Error and Magnitude Estimation Statistics and Probability Collection of Data Organization and Display of Data Analysis of Data Predictions from Data Probability In line with the Children First Initiative Reform, this course’s agenda focuses teachers on improving teaching and learning in mathematics and content area literacy. The course will arm schools with the necessary resources and support to improve instruction to ensure that students have the opportunity to fulfill their highest potential through improved teaching techniques that respond to students’ current needs. (Broken down by National Standards) In Mathematics (using Technology and applied to General Science) Standard 1: Analysis, Inquiry, and Design Students will use mathematical analysis, scientific inquiry, and engineering design, as appropriate, to pose questions, seek answers, and develop solutions. Standard 2: Information Systems Students will access, generate, process, and transfer information using appropriate technologies. Standard 3: Mathematics Students will understand mathematics and become mathematically confident by communicating and reasoning mathematically, by applying mathematics in real-world settings, and by solving problems through the integrated study of number systems, geometry, algebra, data analysis, probability, and trigonometry. Standard 4: Science Students will understand and apply scientific concepts, principles, and theories pertaining to the physical setting and living environment and recognize the historical development of ideas in science. Standard 5: Technology Students will apply technological knowledge and skills to design, construct, use, and evaluate products and systems to satisfy human and environmental needs. Standard 6: Interconnectedness: Common Themes Students will understand the relationships and common themes that connect mathematics, science, and technology and apply the themes to these and other areas of learning. Standard 7: Interdisciplinary Problem Solving 5 Students will apply the knowledge and thinking skills of mathematics, science, and technology to address real-life problems and make informed decisions. GRADE 4 Problem Solving Strand Students will build new mathematical knowledge through problem solving. 4.PS.2 Understand that some ways of representing a problem are more helpful than others 4.PS.3 Interpret information correctly, identify the problem, and generate possible solutions Students will solve problems that arise in mathematics and in other contexts. 4.PS.4 Act out or model with manipulatives activities involving mathematical content from literature 4.PS.5 Formulate problems and solutions from everyday situations 4.PS.6 Translate from a picture/diagram to a numeric expression 4.PS.7 Represent problem situations in oral, written, concrete, pictorial, and graphical forms 4.PS.8 Select an appropriate representation of a problem Students will apply and adapt a variety of appropriate strategies to solve problems. 4.PS.9 Use trial and error to solve problems 4.PS.16 Analyze problems by identifying relationships 4.PS.18 Analyze problems by observing patterns 4.PS.19 State a problem in their own words Students will monitor and reflect on the process of mathematical problem solving. 4.PS.22 Discuss the efficiency of different representations of a problem 4.PS.23 Verify results of a problem 4.PS.24 Recognize invalid approaches 4.PS.25 Determine whether a solution is reasonable in the context of the original problem Reasoning and Proof Strand Students will recognize reasoning and proof as fundamental aspects of mathematics. 4.RP.1 Use representations to support mathematical ideas Students will analyze and evaluate the mathematical thinking and strategies of others. 4.CM.7 Restate mathematical solutions shared by other students 4.CM.8 Consider strategies used and solutions found in relation to their own work Students will use the language of mathematics to express mathematical ideas precisely. 4.CM.9 Increase their use of mathematical vocabulary and language when communicating with others 4.CM.10 Describe objects, relationships, solutions, and rationale using appropriate vocabulary 4.CM.11 Decode and comprehend mathematical visuals and symbols to construct meaning Connections Strand Students will recognize and use connections among mathematical ideas. 4.CN.1 Recognize, understand, and make connections in their everyday experiences to mathematical ideas 4.CN.2 Compare and contrast mathematical ideas 6 4.CN.3 Connect and apply mathematical information to solve problems Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 4.CN.4 Understand multiple representations and how they are related 4.CN.5 Model situations with objects and representations and be able to make observations Students will recognize and apply mathematics in contexts outside of mathematics. 4.CN.6 Recognize the presence of mathematics in their daily lives 4.CN.7 Apply mathematics to solve problems that develop outside of mathematics 4.CN.8 Recognize and apply mathematics to other disciplines Representation Strand Students will create and use representations to organize, record, and communicate mathematical ideas. 4.R.1 Use verbal and written language, physical models, drawing charts, graphs, tables, symbols, and equations as representations 4.R.2 Share mental images of mathematical ideas and understandings 4.R.3 Recognize and use external mathematical representations 4.R.4 Use standard and nonstandard representations with accuracy and detail Students will select, apply, and translate among mathematical representations to solve problems. 4.R.5 Understand similarities and differences in representations 4.R.6 Connect mathematical representations with problem solving 4.R.7 Construct effective representations to solve problems Students will use representations to model and interpret physical, social, and mathematical phenomena. 4.R.8 Use mathematics to show and understand physical phenomena (e.g., estimate and represent the number of apples in a tree) 4.R.9 Use mathematics to show and understand social phenomena (e.g., determine the number of buses required for a field trip) 4.R.10 Use mathematics to show and understand mathematical phenomena (e.g., use a multiplication grid to solve odd and even number problems) Number Sense and Operations Strand Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems. Number Systems 4.N.1 Skip count by 1,000’s 4.N.2 Read and write whole numbers to 10,000 4.N.3 Compare and order numbers to 10,000 4.N.4 Understand the place value structure of the base ten number system: 10 ones = 1 ten 10 tens = 1 hundred 10 hundreds = 1 thousand 10 thousands = 1 ten thousand 4.N.5 Recognize equivalent representations for numbers up to four digits and generate them by decomposing and composing numbers 7 4.N.6 Understand, use, and explain the associative property of multiplication 4.N.7 Develop an understanding of fractions as locations on number lines and as divisions of whole numbers 4.N.8 Recognize and generate equivalent fractions (halves, fourths, thirds, fifths, sixths, and tenths) using manipulatives, visual models, and illustrations 4.N.9 Use concrete materials and visual models to compare and order unit fractions or fractions with the same denominator (with and without the use of a number line) 4.N.10 Develop an understanding of decimals as part of a whole 4.N.11 Read and write decimals to hundredths, using money as a context 4.N.12 Use concrete materials and visual models to compare and order decimals (less than 1) to the hundredths place in the context of money Number Theory 4.N.13 Develop an understanding of the properties of odd/even numbers as a result of multiplication Students will understand meanings of operations and procedures, and how they relate to one another. Operations 4.N.14 Use a variety of strategies to add and subtract numbers up to 10,000 4.N.15 Select appropriate computational and operational methods to solve problems 4.N.16 Understand various meanings of multiplication and division 4.N.17 Use multiplication and division as inverse operations to solve problems 4.N.18 Use a variety of strategies to multiply two-digit numbers by one-digit numbers (with and without regrouping) 4.N.19 Use a variety of strategies to multiply two-digit numbers by two-digit numbers (with and without regrouping) 4.N.20 Develop fluency in multiplying and dividing multiples of 10 and 100 up to 1,000 4.N.21 Use a variety of strategies to divide two-digit dividends by one-digit divisors (with and without remainders) 4.N.22 Interpret the meaning of remainders 4.N.23 Add and subtract proper fractions with common denominators 4.N.24 Express decimals as an equivalent form of fractions to tenths and hundredths 4.N.25 Add and subtract decimals to tenths and hundredths using a hundreds chart Students will compute accurately and make reasonable estimates. Estimation 4.N.26 Round numbers less than 1,000 to the nearest tens and hundreds 4.N.27 Check reasonableness of an answer by using estimation Algebra Strand Students will represent and analyze algebraically a wide variety of problem solving situations. Variables and Expressions: 4.A.1 Evaluate and express relationships using open sentences with one Students will perform algebraic procedures accurately. Equations and Inequalities 4.A.2 Use the symbols <, >, =, and ‚(with and without the use of a number line) to compare whole numbers and unit fractions and decimals (up to hundredths) 4.A.3 Find the value or values that will make an open sentence true, if it contains < or > 8 Students will recognize, use, and represent algebraically patterns, relations, and functions. Patterns, Relations, and Functions 4.A.5 Analyze a pattern or a whole-number function and state the rule, given a table or an input/output box GRADE 5 Problem Solving Strand Students will build new mathematical knowledge through problem solving. 5.PS.2 Understand that some ways of representing a problem are more efficient than others 5.PS.3 Interpret information correctly, identify the problem, and generate possible strategies and solutions Students will solve problems that arise in mathematics and in other contexts. 5.PS.4 Act out or model with manipulatives activities involving mathematical content from literature 5.PS.5 Formulate problems and solutions from everyday situations 5.PS.6 Translate from a picture/diagram to a numeric expression 5.PS.7 Represent problem situations verbally, numerically, algebraically, and/or graphically 5.PS.8 Select an appropriate representation of a problem Students will apply and adapt a variety of appropriate strategies to solve problems. 5.PS.10 Work in collaboration with others to solve problems 5.PS.11 Translate from a picture/diagram to a number or symbolic expression 5.PS.12 Use trial and error and the process of elimination to solve problems 5.PS.13 Model problems with pictures/diagrams or physical objects 5.PS.14 Analyze problems by observing patterns 5.PS.15 Make organized lists or charts to solve numerical problems Students will monitor and reflect on the process of mathematical problem solving. 5.PS.16 Discuss with peers to understand a problem situation 5.PS.18 Determine the efficiency of different representations of a problem 5.PS.19 Differentiate between valid and invalid approaches 5.PS.21 Explain the methods and reasoning behind the problem solving strategies used 5.PS.22 Discuss whether a solution is reasonable in the context of the original problem 5.PS.23 Verify results of a problem Reasoning and Proof Strand Students will recognize reasoning and proof as fundamental aspects of mathematics. 5.RP.1 Recognize that mathematical ideas can be supported using a variety of strategies 5.RP.2 Understand that mathematical statements can be supported, using models, facts, and relationships to explain their thinking Students will make and investigate mathematical conjectures. 5.RP.3 Investigate conjectures, using arguments and appropriate mathematical terms 5.RP.4 Make and evaluate conjectures, using a variety of strategies Students will develop and evaluate mathematical arguments and proofs. 9 5.RP.5 Justify general claims or conjectures, using manipulatives, models, expressions, and mathematical relationships 5.RP.6 Develop and explain an argument verbally, numerically, and/or graphically 5.RP.7 Verify claims other students make, using examples and counterexamples when appropriate Communication Strand Students will organize and consolidate their mathematical thinking through communication. 5.CM.1 Provide an organized thought process that is correct, complete, coherent, and clear 5.CM.2 Explain a rationale for strategy selection 5.CM.3 Organize and accurately label work Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others. 5.CM.4 Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form 5.CN.6 Recognize and provide examples of the presence of mathematics in their daily lives 5.CN.7 Apply mathematics to problem situations that develop outside of mathematics 5.CN.9 Recognize and apply mathematics to other disciplines and areas of interest Representation Strand Students will create and use representations to organize, record, and communicate mathematical ideas. 5.R.1 Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations 5.R.2 Explain, describe, and defend mathematical ideas using representations 5.R.3 Read, interpret, and extend external models 5.R.4 Use standard and nonstandard representations with accuracy and detail Students will select, apply, and translate among mathematical representations to solve problems. 5.R.5 Use representations to explore problem situations 5.R.6 Investigate relationships between different representations and their impact on a given problem Students will use representations to model and interpret physical, social, and mathematical phenomena. 5.R.9 Use mathematics to show and understand mathematical phenomena (e.g., find the missing value that makes the equation true: (3 + 4) + 5 = 3 + (4 + ___ ) Number Sense and Operations Strand Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems. Number Systems 5.N.1 Read and write whole numbers to millions 5.N.2 Compare and order numbers to millions 5.N.3 Understand the place value structure of the base ten number system 10 ones = 1 ten 10 tens = 1 hundred 10 hundreds = 1 thousand 10 thousands = 1 ten thousand 10 10 ten thousands = 1 hundred thousand 10 hundred thousands = 1 million 5.N.4 Create equivalent fractions, given a fraction 5.N.5 Compare and order fractions including unlike denominators (with and without the use of a number line) Note: Commonly used fractions such as those that might be indicated on ruler, measuring cup, etc. 5.N.6 Understand the concept of ratio 5.N.7 Express ratios in different forms 5.N.8 Read, write, and order decimals to thousandths 5.N.9 Compare fractions using <, >, or = 5.N.10 Compare decimals using <, >, or = 5.N.11 Understand that percent means part of 100, and write percents as fractions and decimals Number Theory 5.N.12 Recognize that some numbers are only divisible by one and themselves (prime) and others have multiple divisors (composite) 5.N.13 Calculate multiples of a whole number and the least common multiple of two numbers 5.N.14 Identify the factors of a given number 5.N.15 Find the common factors and the greatest common factor of two numbers Students will understand meanings of operations and procedures, and how they relate to one another. Operations 5.N.16 Use a variety of strategies to multiply three-digit by three-digit numbers Note: Multiplication by anything greater than a three-digit multiplier/ multiplicand should be done using technology. 5.N.17 Use a variety of strategies to divide three-digit numbers by one- and two-digit numbers Note: Division by anything greater than a two-digit divisor should be done using technology. 5.N.18 Evaluate an arithmetic expression using order of operations including multiplication, division, addition, subtraction and parentheses 5.N.19 Simplify fractions to lowest terms 5.N.20 Convert improper fractions to mixed numbers, and mixed numbers to improper fractions 5.N.21 Use a variety of strategies to add and subtract fractions with like denominators 5.N.22 Add and subtract mixed numbers with like denominators 5.N.23 Use a variety of strategies to add, subtract, multiply, and divide decimals to thousandths Students will compute accurately and make reasonable estimates. Estimation 5.N.24 Round numbers to the nearest hundredth and up to 10,000 5.N.25 Estimate sums and differences of fractions with like denominators 5.N.26 Estimate sums, differences, products, and quotients of decimals 5.N.27 Justify the reasonableness of answers using estimation GRADE 6 Problem Solving Strand Students will build new mathematical knowledge through problem solving. 6.PS.2 Understand that some ways of representing a problem are more efficient than others 6.PS.3 Interpret information correctly, identify the problem, and generate possible strategies and solutions Students will solve problems that arise in mathematics and in other contexts. 11 6.PS.4 Act out or model with manipulatives activities involving mathematical content from literature 6.PS.5 Formulate problems and solutions from everyday situations 6.PS.6 Translate from a picture/diagram to a numeric expression 6.PS.7 Represent problem situations verbally, numerically, algebraically, and/or graphically 6.PS.8 Select an appropriate representation of a problem Students will apply and adapt a variety of appropriate strategies to solve problems. 6.PS.10 Work in collaboration with others to solve problems 6.PS.11 Translate from a picture/diagram to a number or symbolic expression 6.PS.12 Use trial and error and the process of elimination to solve problems 6.PS.13 Model problems with pictures/diagrams or physical objects 6.PS.14 Analyze problems by observing patterns 6.PS.15 Make organized lists or charts to solve numerical problems Students will monitor and reflect on the process of mathematical problem solving. 6.PS.16 Discuss with peers to understand a problem situation 6.PS.17 Determine what information is needed to solve problem 6.PS.18 Determine the efficiency of different representations of a problem 6.PS.19 Differentiate between valid and invalid approaches 6.PS.21 Explain the methods and reasoning behind the problem solving strategies used 6.PS.22 Discuss whether a solution is reasonable in the context of the original problem 6.PS.23 Verify results of a problem Reasoning and Proof Strand Students will recognize reasoning and proof as fundamental aspects of mathematics. 6.RP.1 Recognize that mathematical ideas can be supported using a variety of strategies 6.RP.2 Understand that mathematical statements can be supported, using models, facts, and relationships to explain their thinking Students will make and investigate mathematical conjectures. 6.RP.3 Investigate conjectures, using arguments and appropriate mathematical terms 6.RP.4 Make and evaluate conjectures, using a variety of strategies Students will develop and evaluate mathematical arguments and proofs. 6.RP.5 Justify general claims or conjectures, using manipulatives, models, expressions, and mathematical relationships 6.RP.6 Develop and explain an argument verbally, numerically, algebraically, and/or graphically 6.RP.7 Verify claims other students make, using examples and counterexamples when appropriate Students will select and use various types of reasoning and methods of proof. 6.RP.8 Support an argument through examples/counterexamples and special cases 6.RP.9 Devise ways to verify results Communication Strand Students will organize and consolidate their mathematical thinking through communication. 6.CM.1 Provide an organized thought process that is correct, complete, coherent, and clear 6.CM.2 Explain a rationale for strategy selection 12 6.CM.3 Organize and accurately label work Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others. 6.CM.4 Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models, and symbols in written and verbal form 6.CM.5 Answer clarifying questions from others Students will analyze and evaluate the mathematical thinking and strategies of others. 6.CM.6 Understand mathematical solutions shared by other students 6.CM.7 Raise questions that elicit, extend, or challenge others’ thinking 6.CM.8 Consider strategies used and solutions found by others in relation to their own work Students will use the language of mathematics to express mathematical ideas precisely. 6.CM.9 Increase their use of mathematical vocabulary and language when communicating with others 6.CM.10 Use appropriate vocabulary when describing objects, relationships, mathematical solutions, and rationale 6.CM.11 Decode and comprehend mathematical visuals and symbols to construct meaning Connections Strand Students will recognize and use connections among mathematical ideas. 6.CN.1 Understand and make connections and conjectures in their everyday experiences to mathematical ideas 6.CN.2 Explore and explain the relationship between mathematical ideas 6.CN.3 Connect and apply mathematical information to solve problems Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole. 6.CN.4 Understand multiple representations and how they are related 6.CN.5 Model situations with objects and representations and be able to draw conclusions Students will recognize and apply mathematics in contexts outside of mathematics. 6.CN.6 Recognize and provide examples of the presence of mathematics in their daily lives 6.CN.7 Apply mathematics to problem situations that develop outside of mathematics Representation Strand Students will create and use representations to organize, record, and communicate mathematical ideas. 6.R.1 Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations 6.R.2 Explain, describe, and defend mathematical ideas using representations 6.R.3 Read, interpret, and extend external models 6.R.4 Use standard and nonstandard representations with accuracy and detail Students will select, apply, and translate among mathematical representations to solve problems. 6.R.5 Use representations to explore problem situations 6.R.6 Investigate relationships between different representations and their impact on a given problem Students will use representations to model and interpret physical, social, and mathematical phenomena. 13 6.R.7 Use mathematics to show and understand physical phenomena (e.g., determine the perimeter of a bulletin board) 6.R.8 Use mathematics to show and understand social phenomena (e.g., construct tables to organize data showing book sales) 6.R.9 Use mathematics to show and understand mathematical phenomena (e.g., Find the missing value: (3 + 4) + 5 = 3 + (4 + ___) Number Sense and Operations Strand Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems. Number Systems 6.N.1 Read and write whole numbers to trillions 6.N.2 Define and identify the commutative and associative properties of addition and multiplication 6.N.3 Define and identify the distributive property of multiplication over addition 6.N.4 Define and identify the identity and inverse properties of addition and multiplication 6.N.5 Define and identify the zero property of multiplication 6.N.6 Understand the concept of rate 6.N.7 Express equivalent ratios as a proportion 6.N.8 Distinguish the difference between rate and ratio 6.N.9 Solve proportions using equivalent fractions 6.N.10 Verify the proportionality using the product of the means equals the product of the extremes 6.N.11 Read, write, and identify percents of a whole (0% to 100%) 6.N.12 Solve percent problems involving percent, rate, and base 6.N.13 Define absolute value and determine the absolute value of rational numbers (including positive and negative) 6.N.14 Locate rational numbers on a number line (including positive and negative) 6.N.15 Order rational numbers (including positive and negative) Students will understand meanings of operations and procedures, and how they relate to one another. Operations 6.N.16 Add and subtract fractions with unlike denominators 6.N.17 Multiply and divide fractions with unlike denominators 6.N.18 Add, subtract, multiply, and divide mixed numbers with unlike denominators 6.N.19 Identify the multiplicative inverse (reciprocal) of a number 6.N.20 Represent fractions as terminating or repeating decimals 6.N.21 Find multiple representations of rational numbers (fractions, decimals, and percents 0 to 100) 6.N.22 Evaluate numerical expressions using order of operations (may include exponents of two and three) 6.N.23 Represent repeated multiplication in exponential form 6.N.24 Represent exponential form as repeated multiplication 6.N.25 Evaluate expressions having exponents where the power is an exponent of one, two, or three Students will compute accurately and make reasonable estimates. Estimation 6.N.26 Estimate a percent of quantity (0% to 100%) 6.N.27 Justify the reasonableness of answers using estimation (including rounding) 14 Evidence of Participant Application (Projects, collections of student work, action research reports): There will be at least 13 review and reflection questions to be submitted, not including the midterm and final examinations.) Course Administrators will on average provide feedback to teachers within two days of their submitted work. Teachers will build an online portfolio of responses to questions which will culminate with the final examination and culminating activity/project to demonstrate their overall knowledge gained. Throughout the course, participants will be able to communicate to fellow teachers and Course Administrators through open discussion forum (organized by threaded postings). Teachers will submit a project incorporated in the Final Exam which will demonstrate their mastery of the topic – this will include a sample of revised lesson plans and summation strategies to immediately apply in their classroom teaching. Teachers will have unlimited opportunity to communicate with both KDS Course Administrators who will constantly review work submitted. Further Details: Pre-assessment questions will be administered online prior to viewing the lecture, and post-assessment questions will gauge mastery of the lecture’s contents after viewing the lecture. Both assessment tests will consist of multiple choice (also including True/False) questions as well as free response questions. Students’ participation in the lecture can and will be monitored by course administrators. In order to receive credit, students will: review and understand “Course Objectives,” this includes a Topic Summary, Topic Objectives, Topic Goals, and a Topic Outline) and satisfactorily complete the aforementioned Pre- and Post- Assessment questions for each segment of the video content (as further described in the Course Outline). Students must view Video/Audio lectures in their entirety along with synchronized PowerPoint slides. Educators will participate in interactive discussion forums led by the course administrator. Students will take notes on the lecture using the online notepad provided. Students will have unlimited access to course materials including but not limited to PowerPoint slides, transcripts, speaker biographies, and other topic resources and auxiliary materials. 15 Additional Detail on KDS’s Interactive Platform KDS’s Course Administrators oversee and monitor participants’ progress throughout the course. Objectives and course expectations are presented prior to each of our 20 lectures of 90 minutes each. In addition, the participant completes a Pre-Assessment Question and Answer Section to activate prior knowledge before viewing the accompanying lecture. Following the lecture there is also a Post- Assessment Question and Answer Section that is more specific to the information presented by the speaker. This will reflect mastery of the concepts and content taught in each lecture and also require teachers to apply their knowledge in free response questions, where they will demonstrate how they will utilize the lessons taught in their own questions. In this course, each participant is expected to create or redesign an existing lesson plan, integrating concepts, strategies and activities that address the crucial concepts learned from the lectures. Midway through the course participants will complete a midterm exam and, at the conclusion of the course, a final exam is given. These exams are essay format and will be reviewed and graded by the course administrator, as will the lesson plan. Participation in Discussion Boards, Blogging, Journal Writing, Live Texting, and creating Online Portfolios are treated as Classroom Discussions, work compilation, and note-taking opportunities rather than assessment tools. Our assessment tools consist of Pre- and Post-Assessment questions, midterm and final exam essays, and lesson plan designs. The computer tabulates multiple choice and True/False answers while the course administrator reviews and responds to the open-ended questions, essays and lesson plan designs. This course consists of 20 lectures lasting 90 minutes from experts in the field of education. The course is built around the content presented by the nationally renowned lecturers. All of the content and concepts were taken directly from the work of these presenters. The ideas and strategies addressed in this course go beyond classroom teachers to include auxiliary personnel and could be used for adult education as well. The assessment project would include the creation of one lesson plan design and responding to a midterm and two final exam essay questions. The lesson plan requires participants to create a new or adapt an existing lesson plan using concepts and strategies that will challenge all students while specifically addressing the needs of diverse groups of students. The exam questions would require participants to examine the specific 16 needs of students in their classroom or school and challenge them to apply the concepts presented in this course to address those needs. The information presented by the lecturers specifically address high expectations and differentiation in the classroom. Participants are constantly challenged to apply strategies and activities in their own classrooms. The research is supplied by the KDS Lecturers who are experts in their respective fields. The biographies of these speakers as well as supporting reference materials are included below. The use of technology and performance assessment is an integral part of the teaching-learning process at KDS. Participants are expected to use technology to participate in classroom discussions, download and create documents, view streaming video/audio presentations and slideshows, print resources, and to post answers to coursework and exams. In addition, participants are expected to exhibit growth between their performance on Pre- and Post-Assessment Questions. Participation to demonstrate mastery of concepts illustrated in this lecture series will include (but is not limited to): Examples of student work involving manipulatives; Explanations of the use of manipulatives to teach selected concepts; Examples of student work that provide reasons for selecting mental math, paper and pencil or estimation and a calculator to determine answers to problems; Descriptions of activities developed using inquiry based well sequenced activities for selected topics; Completion of a set of questions reflecting the goals of the course outlined above. Student Learning This course will impact student learning by: Providing teachers with easy to implement, well sequenced activities that promote conceptual understanding; Providing teachers with easy to implement, well sequenced activities that relate concrete understanding to symbolic interpretation; Providing teachers with techniques to assess all students’ understanding of skills and concepts so that lesson can be adjusted to meet student need and expand student understanding; Providing teachers with activities that require pattern recognition and descriptions with respect to operational procedures with whole numbers, fractions and decimals; Providing teachers with activities that provide creative practice with operational skills and concepts; 17 Providing teachers with activities that promote number sense, estimation strategies, and foundational understanding so that students can chose appropriate and efficient strategies to determine solutions; Assessment Instruments (including projects, presentations, collections of work, reports): Pre-assessment questions will be administered online prior to viewing the lecture, and post-assessment questions will gauge mastery of the lecture’s contents after viewing the lecture. Both assessment tests will consist of both multiple choice (also including True/False) questions as well as free response questions. Students’ participation in the lecture can and will be monitored by course administrators. In order to receive credit, students will: review and understand “Course Objectives” and satisfactorily complete the aforementioned Pre- and Post-Assessment questions for each segment of the video content (as further described in the Course Outline). Students must view Video/Audio lectures in their entirety along with synchronized PowerPoint slides. Students will participate in interactive discussion forums led by the course administrator. Students will take notes on the lecture using the online notepad provided. Students will have unlimited access to course materials including but not limited to PowerPoint slides, transcripts, speaker biographies, and other topic resources and auxiliary materials. Administrators will have access to: • Student usage statistics, • Discussion boards, • Teacher blogs, • Note-taking sessions, • Pre-Assessment Q&A, • Post-Assessment Q&A, • Cumulative Coursework File Students will submit assessment questions for a Midterm and Final Exam in the same fashion as submission of assessment questions. 18 COURSE OUTLINE: Lecturer Name(s): Paul Lawrence Title of Course: (Online Course) Discovery-Based Mathematics Course Location: Online (via home, school, library computer, etc.) Instructor’s Telephone #: (212) 809-2969 E-mail: mbfox@kdsi.org, schools@kdsi.org Course Coordinator: Melanie Fox COURSE DESCRIPTION (SEE NEXT PAGE FOR DETAILS) 19 COURSE DESCRIPTION Calendar Topics Goals/ Method of Instruction Text/Readings Objective Session 1: Essential Instructional Strategies for all Math Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours Classrooms – the need for change based techniques that: the topics presented during the 4.N.2 session. Connections to Everyday 4.N.3 Model and connect Mathematics as appropriate NOTE: ALL Concepts of the base 10 number system 4.N.4 manipulatives to SESSIONS using base ten squares, ten-thousand 4.N.5 algorithms. Required Participant Manipulative can be split squares and 1,000,000 charts 5.N.1 Kit into several 5.N.2 Use response devices to sessions, as Compare and order sets of whole numbers 5.N.3 asses understanding Communicator™ 531664 to be using models and discovery 5.N.12 Clean Wipe and Marker determined 5.N.13 Include discussions and Fraction Tiles (525074) by the user Compare and order whole numbers on a 5.N.14 explanations of why 100 Square Geoboards (525042) number line 5.N.15 various strategies are Overhead Base Ten Blocks 5.N.27 attached to the context of (525056) Discuss relationships among multiples of 10 6.N.1 the problem and choice of 100 connecting cubes (10 each of 6.N.2 using mental math, paper each of 10 colors) (530118) Using cubes to classify prime, and composite and pencil or estimation Overhead tiles(525033) numbers All indicators and calculator to determine Overhead pattern blocks (525017) described in answers to problems. Overhead playing cards (525024) Using Arrays to Model Prime and Composite standards Assorted Dice (530176) Numbers listing above Model games that help Communicator Mathematics™ for students master partial Content Series Implementation Using Cubes to Model Prime and Composite Problem sum method and mental RAL Packet: Numbers Solving, math techniques. Samples of classroom activity Reasoning sheets for Sessions 1-12. (A set of Exponential notation and Proof, 9 student booklet a total of Connections, approx400 pages of activities).) Prime Factorization Communicati on, All items available from Greatest Common Factors Representati EAIEducation.com ($99.95) on, Least Common Multiples Fraction Capable Calculator (not included in kit may be ordered Divisibility Rules separately) Suggested materials for classroom implementation not required for course enrollment 20 Communicator™ Clearboard classroom kit EAI Education.com Communicator Mathematics™ Content Series Whole Number Teacher Guide series Communicator Mathematics™ Content Series Fraction and Decimal Teacher Guide Series Session 2: Exploring Negative numbers, scientific Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours notation and order of operations based techniques that: the topics presented during the 5.N.18 session. And connections to Introducing and representing negative 5.N.27 Model and connect Everyday Mathematics as numbers 5.N.13 manipulatives to appropriate Comparing Integers – Vertical Number Line 5.N.14 algorithms. Comparing Integers – Horizontal Number 5.N.15 See listing from session 1 Lines 6.N.22 Use response devices to Ordering Integers 6.N.236.N.24 asses understanding 6.N.25 Concepts of Scientific Notation 6.N.27 Include discussions and Standard form to Exponential and Scientific explanations of why Notation All indicators various strategies are Scientific notation to Standard Form described in attached to the context of Transferring between scientific and standard standards the problem and choice of form listing above using mental math, paper for and pencil or estimation Order of Operations Problem and calculator to determine Multiply and divide from left to right Solving, answers to problems. Parentheses first Reasoning Exponents and Proof, Model games that help Adding and subtracting from left to right Connections, students master partial Discovering and applying PEMDAS Communicati sum method and mental on, math techniques. Representati on Session 3: Addition and subtraction facts with Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours connecting cubes based techniques that: the topics presented during the 4.N.14 session and connections to 21 Double digit addition and subtraction with 4.N.15 Model and connect Everyday Mathematics as base ten blocks 4.N.27 manipulatives to appropriate. 4.A.2 algorithms. Double digit addition and subtraction with 4.A.3 See listing from session 1 hundred charts 5.N.27 Use response devices to All indicators asses understanding Attaching concrete modeling to symbolic described in algorithms standards Include discussions and listing above explanations of why Mental math and problem solving activities for various strategies are with basic addition and subtraction facts up Problem attached to the context of to double digits Solving, the problem and choice of Reasoning using mental math, paper Implementing games that provide conceptual and Proof, and pencil or estimation practice of addition Connections, and calculator to determine Communicati answers to problems. Alternative strategies for multi-digit addition on, and subtraction. Same change rule, trade Representati Model games that help first, counting up, partial sum method, on students master partial opposite change rule, column addition, sum method and mental traditional algorithm math techniques. Estimating answers to multi-digit addition and subtraction problems Determining and applying efficient strategies to add and subtract mixed sets of multi-digit addition and subtraction problems Session 4: Developing conceptual understanding of Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours multiplication through arrays, groups of 4.N.16 based techniques that: the topics presented during the things, and repeated addition. 4.N.17 session and connections to 4.N.18 Model and connect Everyday Mathematics as Creating student multiplication books for 4.N.20 manipulatives to appropriate. each times table using arrays, groups of 4.N.27 algorithms. things and repeated addition 4.A.5 See listing from session 1 5.N.27 Use response devices to Creating a multiplication table, analyzing 6.N.2 asses understanding patterns and creating a strategies to 6.N.27 memorize facts based on these patterns All indicators Include discussions and described in explanations of why 22 Simulating the standard multiplication table standards various strategies are on geoboards listing above attached to the context of for the problem and choice of Using response devices to master Problem using mental math, paper multiplication facts through the application of Solving, and pencil or estimation arrays and fact recall Reasoning and calculator to determine and Proof, answers to problems. Games that enhance conceptual Connections, understanding of multiplication and recall of Communicati Model games that help facts on, students master partial Representati sum method and mental Use input/output tables to review and on math techniques. master basic multiplication facts Developing conceptual understanding of basic division through groups, fair sharing, arrays, and the inverse operation of multiplication Using connecting cubes and geoboards to model division facts and real-life applications Using base ten blocks and grouping to extend multiplication facts to include multiples of 10 Using arrays and base ten blocks to understand single digit times double digit multiplication Attaching the partial product algorithm to base ten solutions of single digit times double digit numbers Use mental math techniques to determine products of single digit times double digit numbers 23 Session 5: Developing conceptual understanding of Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours double digit times double digit multiplication based techniques that: the topics presented during the using base ten block and area/array models 4.N.19 session. And connections to 4.N.27 Model and connect Everyday Mathematics as 4.N.21 manipulatives to appropriate Attaching the partial product algorithm to 4.N.22 algorithms. base ten solutions of double digit times 5.N.16 See listing from session 1 double digit numbers 5.N.17 Use response devices to 5.N.27 asses understanding Developing conceptual understanding of 6.N.27 multi-digit multiplication using base ten Include discussions and block and area/array models All indicators explanations of why described in various strategies are standards attached to the context of Attaching the partial product algorithm to listing above the problem and choice of area model solutions of multi-digit for using mental math, paper multiplication problems. Problem and pencil or estimation Solving, and calculator to determine Reasoning answers to problems. Estimate and validate estimations to multi- and Proof, digit multiplication problems Connections, Model games that help Communicati students master partial Determining and applying efficient strategies on, sum method and mental to multiply sets of single and multi-digit Representati math techniques. problems on Developing conceptual understanding of partial quotients with single digit divisors Developing conceptual understanding of partial quotients with double digit divisors Connecting conceptual understanding of division of single and double digit divisors to fair sharing strategies Estimate and validate estimations to quotients that include problems with multi- digit divisors 24 Determining and applying efficient strategies to divide sets of division problems Determining and applying efficient strategies to add, subtract, multiply or divide mixed sets of multiplication problems. Play games that will enhance number sense and estimation and skill mastery of whole number operations (Three digit or four digit fun.) Session 6: Concepts of fractions and decimals Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours Equivalent fractions based techniques that: the topics presented during the 4.N.7 session. And connections to Show fractions using linear, area, and group 4.N.8 Model and connect Everyday Mathematics as models 4.N.9 manipulatives to appropriate 4.N.10 algorithms. Create a unit stick using construction paper 4.N.11 See listing from session 1 and scotch tape 4.N.11 Use response devices to 4.N.12 asses understanding Understand fractional relationships and 4.N.24 equivalencies using rulers 4.N.27 Include discussions and 5.N.6 explanations of why Understand fractional relationships and 5.N.7 various strategies are equivalencies using fraction tiles and 5.N.8 attached to the context of equivalence templates 5.N.9 the problem and choice of 5.N.10 using mental math, paper Discover and apply the algorithm to 5.N.11 and pencil or estimation determine fraction equivalencies 5.N.19 and calculator to determine 5.N.20 answers to problems. Discover and apply decimal approximations 5.N.27 and equivalencies 6.N.6 Model games that help 6.N.7 students master partial Order and compare sets of fractions and 6.N.8 sum method and mental decimals using number lines 6.N.20 math techniques. 25 6.N.21 Order and compare sets of fractions and 6.N.27 decimals using mental math. All indicators described in See improper fractions and mixed numbers standards as sums listing above for Problem Solving, Reasoning and Proof, Connections, Communicati on, Representati on Session 7: Addition and subtraction of fractions with Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours same denominators and compatible based techniques that: the topics presented during the denominators 4.N.23 session. And connections to 4.N.27 Model and connect Everyday Mathematics as Define compatible denominators such that 5.N.21 manipulatives to appropriate one denominator is a multiple of the other 5.N.22 algorithms. 5.N.25 See listing from session 1 Add and subtract sequenced sets of fractions 5.N.27 Use response devices to with like denominators using fraction tiles, 6.N.27 asses understanding rulers and equivalence charts All indicators described in Include discussions and Use mental math to add and subtract standards explanations of why fractions with like denominators using listing above various strategies are mental math. for attached to the context of Problem the problem and choice of Add and subtract sequenced sets of fractions Solving, using mental math, paper with compatible denominators using fraction Reasoning and pencil or estimation tiles, rulers and equivalence charts and Proof, and calculator to determine Connections, answers to problems. Use mental math to add and subtract Communicati fractions with compatible denominators using on, Model games that help mental math. Representati students master partial on sum method and mental Play games that will enhance the math techniques. 26 understanding of the addition and subtraction of fractions with like and compatible denominators. Session 8: Addition and subtraction of fractions with Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours non-compatible and overlapping based techniques that: the topics presented during the denominators 6.N.16 session. And connections to 2.N.27 Model and connect Everyday Mathematics as Define non-compatible and overlapping All indicators manipulatives to appropriate denominators described in algorithms. standards See listing from session 1 Add and subtract sequenced sets of fractions listing above Use response devices to with non-compatible denominators using for asses understanding fraction tiles, rulers and equivalence charts Problem Solving, Include discussions and Connect procedural understanding to Reasoning explanations of why traditional algorithms for addition and and Proof, various strategies are subtraction of non-compatible denominators Connections, attached to the context of Communicati the problem and choice of Add and subtract sequenced sets of fractions on, using mental math, paper with overlapping denominators using fraction Representati and pencil or estimation tiles, rulers and equivalence charts on and calculator to determine answers to problems. Connect procedural understanding to traditional algorithms for addition and Model games that help subtraction of overlapping denominators students master partial sum method and mental Estimate answers to addition and subtraction math techniques. fraction problems Use appropriate calculator keystrokes to determine sums and differences of fraction problems Determine and apply efficient methods and strategies to add and subtract fractions by using mental math, paper and pencil, or estimation and a calculator based on the context of the problem Play games that will enhance the 27 understanding of the addition and subtraction of fractions Session 9: Multiplication of fractions Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours 6.N.17 based techniques that: the topics presented during the Using fraction tiles to understand the 6.N.27 session. And connections to multiplication of proper fractions times Model and connect Everyday Mathematics as proper fractions All indicators manipulatives to appropriate described in algorithms. Using an area model to understand the standards See listing from session 1 multiplication of proper fractions times listing above Use response devices to proper fractions for asses understanding Problem Using fraction tiles to understand the Solving, Include discussions and multiplication of proper fractions times whole Reasoning explanations of why numbers and mixed numbers and Proof, various strategies are Connections, attached to the context of Using an area model to understand the Communicati the problem and choice of multiplication of proper fractions times whole on, using mental math, paper numbers and mixed numbers Representati and pencil or estimation Discovering the algorithm to multiply proper on and calculator to determine fractions and proper fractions times whole answers to problems. numbers Model games that help Using an area model to understand the students master partial multiplication of mixed numbers times mixed sum method and mental numbers math techniques. Developing an algorithm to represent the area model of multiplying mixed numbers times mixed numbers Estimating products of mixed numbers times mixed numbers. Use appropriate keystrokes on a calculator to determine the product of two fractions Determine and apply efficient methods and strategies to multiply fractions by using 28 mental math, paper and pencil, or estimation and a calculator based on the context of the problem Play games that will enhance the understanding of the multiplication of fractions Session 10: Division of fractions Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours 6.N.4 based techniques that: the topics presented during the Use fraction tiles to determine the quotients 6.N.17 session. And connections to of whole numbers divided by proper fractions 6.N.18 Model and connect Everyday Mathematics as 6.N.19 manipulatives to appropriate Use rulers to determine the quotients of 6.N.27 algorithms. whole numbers divided by proper fractions All indicators See listing from session 1 described in Use response devices to Use fraction tiles to determine the quotients standards asses understanding of proper fractions divided by proper listing above fractions for Include discussions and Problem explanations of why Use rulers to determine the quotients of Solving, various strategies are proper fractions divided by proper fractions Reasoning attached to the context of and Proof, the problem and choice of Use discovery to determine the standard Connections, using mental math, paper fraction division algorithm Communicati and pencil or estimation on, and calculator to determine Use fraction tiles to determine the quotients Representati answers to problems. of mixed numbers divided by proper on fractions Model games that help Estimating products of mixed numbers times students master partial mixed numbers. sum method and mental math techniques. Use appropriate keystrokes on a calculator to determine the quotient of two fractions Determine and apply efficient methods and strategies to find the quotient fractions by using mental math, paper and pencil, or estimation and a calculator based on the context of the problem 29 Determine and apply efficient methods and strategies to determine answers to mixed sets of fraction problems using mental math, paper and pencil, or estimation and a calculator based on the context of the problem Session 11: Addition and subtraction of decimals Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours 5.N.23 based techniques that: the topics presented during the Use linear and area models to add and 5.N.27 session. And connections to subtract tenths 6.N.27 Model and connect Everyday Mathematics as All indicators manipulatives to appropriate Use linear and area models to add and described in algorithms. subtract hundredths standards See listing from session 1 listing above Use response devices to Use area models to add and subtract for asses understanding thousandths Problem Solving, Include discussions and Use the partial sum method to add and Reasoning explanations of why subtract sets of decimals and Proof, various strategies are Connections, attached to the context of Estimate sums and differences of various Communicati the problem and choice of decimal addition and subtraction problems on, using mental math, paper Representati and pencil or estimation Determine and apply efficient methods and on and calculator to determine strategies to find the s sums and differences answers to problems. of various decimal addition and subtraction problems by using mental math, paper and Model games that help pencil, or estimation and a calculator based students master partial on the context of the problem sum method and mental math techniques. Session 12: Multiplication and Division of decimals Goals 1-7 Video showing inquiry Online handouts that reflect each 2.5 hours based techniques that: the topics presented during the Review of decimal concepts 5.N.26 session. And connections to 5.N.27 Model and connect Everyday Mathematics as Model products less than 1 using unit 6.N.27 manipulatives to appropriate squares All indicators algorithms. Model products greater than 1 using an area described in See listing from session 1 model standards Use response devices to Transfer to the partial product method listing above asses understanding 30 Use number sense and estimation to develop for Problem multiplication algorithm Solving, Include discussions and Reasoning explanations of why Use linear models to determine quotients to and Proof, various strategies are whole number divisors with decimal Connections, attached to the context of dividends Communicati the problem and choice of Use function machines to determine on, using mental math, paper quotients to selected decimal division Representati and pencil or estimation problems on and calculator to determine Use number sense and estimation to answers to problems. determine quotients Model games that help Determine and apply efficient methods and students master partial strategies to find the quotients off various sum method and mental decimal division problems by using mental math techniques. math, paper and pencil, or estimation and a calculator based on the context of the Video showing inquiry problem based techniques that: Determine and apply efficient methods and Model and connect strategies to find the solutions to mixed sets manipulatives to of decimal problems using mental math, algorithms. paper and pencil, or estimation and a calculator based on the context of the Use response devices to problem asses understanding Choosing efficient methods and strategies to Include discussions and solve mixed sets of operations with whole explanations of why numbers, fractions, and decimals by various strategies are choosing from mental math, paper and attached to the context of pencil, or estimation and a calculator. the problem and choice of using mental math, paper and pencil or estimation Revisit essential instructional strategies and calculator to determine outlined in the first session and give answers to problems. examples from the course that modeled these essential strategies Model games that help students master partial Discuss effective ways to connect alternative sum method and mental strategies and manipulatives Board adopted math techniques. curriculum and texts. 31 Revisit and discuss ways in which using geoboards, base ten blocks, connecting cubes and discovery learning has changed the culture in the way in which you have taught mathematics during this course. TOTAL LECTURE HOURS = 30 Part I = 15 hours, Part II = 15 hours 32

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posted: | 7/20/2010 |

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