KDS COURSE #6 Discovery-Based Mathematics

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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;
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       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
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       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


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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
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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
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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 >
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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.
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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
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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.
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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)



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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




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