After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201 Helaine Schwartz, Director [email protected]Revised 11.2011 his Course Syllabus Template All Courses = 36 Hours; minimum 6 sessions Summer Term –July 2, 2012 to August 17, 2012 Please complete a full course syllabus using this format. The number of sessions held will depend on how you allocate the 36 hours. This syllabus will be uploaded to the ASPDP web site. Please be sure it is in a word or PDF document format. Title of Course: Discovery-Based Mathematics Course Code: #P12-86SS12 Course Location: Online, www.kdsi.org/NYC Instructor’s Name: Dr. Diane Moroff / Presenter: Paul Lawrence Instructor’s Telephone #: 1-800-728-0032 E-mail: [email protected]Course Begins: July 2, 2012 Course Ends: August 17, 2012 Total Hours: 53 hours Course Description These numbers don’t lie. Test scores soar when students have a true understanding of number sense. Discovery-Based Mathematics is a hands-on, inquiry-based approach to math that grounds student knowledge firmly in number sense and then develops conceptual understanding, so that students can do double-digit computation … in their heads! Presenter Paul Lawrence leads educators through easy-to-implement, well-sequenced activities that build foundational and conceptual understanding in real, whole, and negative numbers and addition and subtraction of whole numbers. Using a variety of manipulatives, Lawrence demonstrates the importance of hands-on discovery-based learning to move students from concrete to iconic to symbolic representations, before introducing procedures. His methods address kinesthetic and visual as well as abstract learning styles. Educators follow along, using the materials in the Discovery-Based Math Manipulatives Kit, as they do the same activities workshop participants do. Educators learn techniques, activities, and games to assess students’ skills and concept understanding, so that lessons can be adjusted to meet the needs of all learners. Required Discovery-Based Math Manipulatives Kit ($69) The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,” along with many manipulatives for hands-on activities. (The kit materials are packaged in a convenient carrying case.) NOTE: Midterms may be submitted anytime but are due no later than 2 weeks after the final registration date. Calendar Session # 1 Date: self-paced Time: self-paced
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Course Syllabus Template All Courses = 36 Hours; minimum 6
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After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Employ math manipulatives in instruction
Use the Communicator™ and discovery templates
Use discovery-based methods to help students understand the base ten number system
Guide students to illustrate, compare, and order sets of whole numbers
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 3 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Classifying, Ordering and Exploring Real Numbers: Part 2
Paul Lawrence considers how discovery-based learning can guide students to master essential math
concepts including: rounding, ordering, exponential notation, prime numbers, and composite
numbers. Lawrence models using variety of math manipulatives, such as math cubes, geoboards,
and arrays, as well as templates to help students visualize numbers and other math concepts and
develop critical higher order thinking and problem solving skills.
Objectives: Specify instructional goals and standards for each session.
Goals:
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 4 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Classifying, Ordering and Exploring Real Numbers, Part 3
In this unit, Paul Lawrence introduces the idea of a function machine. He explores ways function
machines can be used for factorization, finding the greatest common factor, and finding the least
common multiples. Lawrence also demonstrates an inquiry-based approach that leads students to
discover divisibility rules.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this session, educators will know:
What a function machine is and how it can be used to teach essential math concepts
Strategies to help students discover and understand divisibility
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
Students will:
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
New York State Learning Standards
The course satisfies these Mathematics, Science, and Technology standards:
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this session, educators will apply the following skills:
Use function machines to teach essential math concepts
Employ strategies that can be used to help students understand divisibility
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 5 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Exploring Negative Numbers, Scientific Notation and Order of Operations
In this unit, Paul Lawrence explores how teachers can help students understand comparisons
between negative and positive numbers through their familiarity with vertical number lines (i.e.,
thermometers). He demonstrates how scientific notation can be taught and used in calculations
without introducing the idea of moving the decimal point. Finally, he addresses order of operations
and provides a model for leading students to discover the correct order without relying on the
acronym PEMDAS (Please Excuse My Dear Aunt Sally). He emphasizes the point that calculators
can be effective tools to help students understand and explore concepts when used during concept-
discovery problem solving.
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 6 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Addition and Subtraction of Whole Numbers: Part 1
Paul Lawrence explores how teachers can help student understand basic addition and subtraction of
whole numbers. Using connecting cubes, geoboards, hundred blocks and charts, Lawrence
demonstrates how to extend fact practice to include algebraic thinking skills long before the
standard introduction to the algorithms. Lawrence explains the value of teaching subtraction as
counting on as a means for practicing making change and calculating elapsed time.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this unit, educators will know:
Why manipulatives should be used for understanding of basic addition and subtraction facts
The sequence of steps in developing understanding to the symbolic stage
The value of teaching subtraction as counting on
How to use flats, rods, and units to add and subtract
Multiple strategies for solving math problems
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
Students will:
Make sense of problems and persevere in solving them
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Use the games “Win a Flat,” “Lose a Flat,” and “Column Addition” to teach addition and
subtraction
Teach multiple strategies for solving math problems
Teach students to estimate answers to math problems
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 8 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Mastering Multiplication and Division Facts: Part 1
In this first of a three units on mastering multiplication and division facts, Lawrence leads teachers
through several exercises to ground student understanding in concrete experiences. He uses Elinor
Pinczes’ One Hundred Hungry Ants, manipulatives, and student-made booklets to make the abstract
personal. Lawrence demonstrates how to use arrays to reinforce concepts of rows and columns and
the commutative property of multiplication. He urges teachers to ensure that students have a firm
grasp of the concept of multiplication before introducing the times tables. In doing so, he carefully
moves from the concrete to the iconic and then the symbolic. Lawrence also stresses the importance
of using more than one approach to reach the same end — i.e., to facilitate students’ mastery of the
concept.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this unit, educators will know:
Why they should use manipulatives to teach multiplication
Why it is important to use multiple strategies when teaching math concepts
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Calendar Session # 9 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Mastering Multiplication and Division Facts: Part 2
In the second of a three-part session on mastering multiplication and division facts, Paul Lawrence
continues to model multiple strategies for teaching multiplication. Having demonstrated using
arrays and groups of things in part 1, Lawrence models the strategy of repeated addition. Once
again, he shows how to use a calculator, this time as function machine for repeated addition to
explore concepts. Lawrence reviews how to have students make their own individual multiplication
booklets. Incorporating higher-order thinking skills, Lawrence introduces using open-ended math
questions and writing about math to further understanding as well as analytic and writing skills.
After students fully understand the concepts, Lawrence offers effective strategies for memorize the
multiplication tables for numbers 1 through 10. He shows how of the 100 facts, there are just 15
“hard” facts to memorize. Finally he demonstrates how teachers can employ geoboards to
simultaneously teach rectangles, areas, and multiplication facts.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this unit, educators will know:
How to use a calculator as a function machine
Why open-ended questions and discussion in math instruction are important
Ways to use geoboards to teach multiple concepts
How to construct practice pages that support conceptual understanding
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
Students will:
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Look for and express regularity in repeated reasoning
New York State Learning Standards
The course satisfies these Mathematics, Science, and Technology standards:
Analysis, Inquiry, and Design
Mathematics
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Program and use a calculator as a function machine for repeated addition
Use open-ended math questions, written answers, and analyzing the answers of others to
promote concept comprehension, writing, and thinking skills
Use geoboards to review or teach multiple concepts
Develop practice pages that support concept understanding with iconic representations of
algorithms
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 10 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Mastering Multiplication and Division Facts: Part 3
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Use games to reinforce instruction in multiplication and division
Use function machines to teach multiplication
Use partial product algorithm
Implement a sequence of steps to teach students mental math with multi-digit multiplication
problems
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 11 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Mastering Multiplication and Division Beyond the Facts: Part 1
This is the first of a two more units on multiplication and division, moving beyond basic facts to
understanding multi-digit computation. Paul Lawrence demonstrates how to use arrays and partial
product methods to do multi-digit multiplication problems. He also introduces ideas and
suggestions for helping students learn to decide when a problem is best solved using pencil and
paper, estimation, or a calculator.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this unit, educators will know:
How to use arrays to teach multi-digit multiplication
How to use the partial product algorithm to teach multi-digit multiplication
When and how to estimate and validate estimations for multi-digit multiplication problems
When and why students should use calculators
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Calendar Session # 12 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Mastering Multiplication and Division Beyond Facts: Part 2
This is the second of a two units that move beyond basic multiplication and division. Lawrence
demonstrates how teachers can use tile templates, play money, and partial quotient and “fair share”
methods to help their student understand and solve multi-digit division problems. Lawrence shares
worksheets from his programs in which students need to apply number sense to decide the best
method for solving problems along with computation practice using different strategies. This unit
concludes with Three Digit Fun, a game for students to practice multiple skills and strategies as
they figure out how to make three digits into problems that equal the given answers.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
How to use templates and acting out to teach multi-digit division
Why multiple strategies should be used to teach multi-digit division
How to apply estimating skills to multi-digit division problem practice
When to have students use calculators
How to use games to practice multiple math skills
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
Students will:
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
New York State Learning Standards
The course satisfies these Mathematics, Science, and Technology standards:
Analysis, Inquiry, and Design
Mathematics
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Use templates and acting out to teach multi-digit division
Teach fair share and partial quotient algorithm to solve division problems
Apply estimating skills to multi-digit division problem practice
Appropriately use calculators with division
Use games to practice multiple math skills
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 13 Date: self-paced Time: self-paced Number of hours for this session: 5 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Discovery-Based Mathematics
Midterm
Paul Lawrence urges teachers to spend much, much more time with manipulatives to cement foundational
concepts before moving to icons, then attaching those icons to symbols.
Take a current lesson plan for one of the topics in the course so far. Prepare discovery, inquiry-based,
hands-on experiences that use manipulatives of your choice to provide concrete and conceptual
understanding before introducing the traditional algorithm.
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
1. Select a math standard or learning objective for one of the topics in the course.
2. Describe a method or device, such as Lawrence’s Communicator™, to check student responses and
understanding at a glance.
3. Identify the manipulatives, templates, games and game pieces, etc., that you will use. Adapt or use
the Discovery Templates in the course handbook, if appropriate.
4. Ask; don’t tell summarizes Paul Lawrence’s inquiry-based approach to math. Develop a series of
questions to take students through the discovery process. Identify the questions that ask students to
summarize, analyze, make conjectures, and generalize by adding (s), (a), (c) or (g) at the end of each
question.
5. Include at least two activities or games to teach the concept in different ways to meet differentiated
learning needs. Identify the instructional need and describe the steps you will use to teach students
the game.
6. Explain how you will transfer students’ concrete experiences to iconic representations.
7. Create a rubric to assess students’ understanding after completing the activities.
Objectives: Specify instructional goals and standards for each session.
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Guest Speakers:
Calendar Session # 14 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Concepts of Fractions and Decimals: Part 1
This is the first of three units on fractions and decimals. Paul Lawrence begins with an overview of
what the units are leading up to by modeling how to guide students to choose which fraction
problems are best done with mental math, paper and pencil, estimation, and calculators. As with
other topics, Lawrence offers teachers several techniques to help students understand fractions,
compare fractions, and estimate, that use both area and linear models. Lawrence then models
making unit sticks to help students see fraction equivalences. He also demonstrates how to use the
unit sticks to give students practice with estimating using nonstandard measurement.
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Use classroom management techniques to support estimating activities
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 15 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Concepts of Fractions and Decimals: Part 2
In the second of three units on fractions and decimals, Paul Lawrence offers teachers several
techniques that they can use to help their students understand simplifying fractions and write
equivalence and inequality statements. He models using a unit stick in conjunction with a giant inch
template that will help students move from area to linear models when comparing fractions. He
further demonstrates how teachers can use simulated rulers to compare fractions and how the
fraction number template can be used to help students compare fifths, thirds, sevenths, and
hundreds as well as halves, fourths, eights, and sixteenths. Finally, he demonstrates how to use a
calculator to simplify fractions after students have understood the concept of simplification.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
Fractional relationships using rulers
How to use rulers to illustrate fractional relationships and at the same time teach students to
read a ruler
How to use the Fraction Stick Template
How to teach students to simplify fractions
When to use a calculator to simplify fractions
Standards
A Framework for Teaching
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Calendar Session # 16 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Concepts of Fractions and Decimals: Part 3
In the third unit on fractions and decimals, Paul Lawrence explores fractions and decimals using
concrete and iconic representations for single units (divided into fractional parts) and groups of
things. He models how to move from fractional equivalences to decimals, including how to use a
calculator to demonstrate the traditional algorithm (divide the numerator by the denominator).
Through a variety of templates with line segments, shapes, and grids, he shows teachers how to
help students visualize decimals and discover how to compare them. Finally he introduces methods
for teaching fractions as groups of things.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
How to use the Fraction Stick template to compare fractions
How to use the Fraction Stick template to teach fraction-to-decimal conversions
When and how to use a calculator to do the fraction-to-decimal procedure
How to use linear, area, and money to model tenths
How to use grids to develop concepts of tenths, hundredths, and thousandths
Templates to use to teach fractions as groups of things
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
Students will:
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
The course satisfies these Mathematics, Science, and Technology standards:
Analysis, Inquiry, and Design
Mathematics
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Compare fractions using the Fraction Stick template
Convert fractions to decimals using the Fraction Stick template
How to use a calculator to do the fraction-to-decimal procedure
Model tenths using linear, area, and money
Develop concepts of tenths, hundredths, and thousandths
Teach fractions as groups of things using 24 and 60 penny templates
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 17 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Addition and Subtraction of Fractions with Same and Compatible Denominators
In this unit, Paul Lawrence focuses on techniques to teach adding and subtracting fractions with the
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this session, educators will apply the following skills:
Use a variety of manipulatives and templates to teach addition and subtraction of fractions
with like and compatible denominators
Transfer concepts from iconic experiences to mental math to solve addition and subtraction
problems involving fractions
Use games to enhance and practice addition and subtraction of fractions with like and
compatible denominators
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 18 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Addition and Subtraction of Fractions with Non-Compatible and Overlapping Denominators
In this unit, Paul Lawrence focuses on adding and subtracting fractions with non-compatible and
overlapping denominators. He develops conceptual understanding through multiple strategies using
Fraction Files and Fraction Sticks and by counting on. He demonstrates the importance of using
number sense to estimate and when and how calculators should be used with fractions in the
classroom. The unit concludes with another “You Decide” page of problems and guidelines to
determine whether problems should be solved using mental math, paper and pencil, or estimation
and a calculator.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
How to use Fraction Tiles and Fraction Sticks to teach addition and subtraction of fractions
with non-compatible and overlapping denominators
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
When and how to use calculators appropriately to solve fraction problems
How to teach subtraction of fractions by counting on
How to use number sense to estimate with fractions
Techniques for determining the best method to solve fraction problems
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
Students will:
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
New York State Learning Standards
The course satisfies these Mathematics, Science, and Technology standards:
Analysis, Inquiry, and Design
Mathematics
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Teach addition and subtraction of fractions with non-compatible and overlapping denominators using Fraction Tiles and Fraction Sticks
Apply number sense to estimate addition and subtraction of fractions
Teach the appropriate use of calculators to solve fraction problems
Teach subtraction of fractions by counting on
Teach strategies to determine whether mental math, paper and pencil, or a calculator should be used to solve problems based on the context of the problem
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 19 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Multiplication of Fractions
Paul Lawrence focuses on techniques that can help students understand the concept of the
multiplication of fractions. Connecting what students already know about multiplication of whole
numbers to fractions, Lawrence models how to use arrays and Fraction Tiles to multiply fractions.
He also demonstrates using iconic drawings to illustrate problems without using manipulatives. As
in other units, Lawrence shows how and when calculators should be used with fractions in the
classroom. This unit covers multiplication of proper fractions times proper fractions, proper
fractions times whole numbers and mixed numbers, and mixed numbers times mixed numbers.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
How to use area models to understand multiplication of fractions
How to use Fractions Tiles to understand multiplication of fractions
How to use hand-drawn models to determine products of proper fractions
How to guide students to discover the algorithm
Why students should learn to estimate products of fractions
When and how to use calculators to determine the product of two fractions
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Calendar Session # 20 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Division of Fractions
As with previous topics, Paul Lawrence begins this unit on division of fractions with concrete
experiences. He models the concept using Fraction Tiles. Lawrence develops division of fractions
from simple to more difficult (proper fractions, proper fractions into whole numbers and mixed
numbers, and mixed numbers into mixed numbers) with lots of practice before introducing the
procedure. He also models estimating answers as the first step to using calculators with division of
fractions. Finally, he offers guidelines for applying number sense to division of fractions to
determine whether problems should be solved using mental math, paper and pencil, or estimation
and a calculator.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
How to use Fraction Tiles to introduce and practice division of fractions
How to develop the standard fraction division algorithm through discovery
How to determine and apply efficient methods and strategies to find quotients by using mental math, paper and pencil, or estimation and a calculator based on the context of the problem
How to apply number sense to estimate answers
How to use a fraction capable calculator to solve fraction problems after estimating
Standards
A Framework for Teaching
3c: Instruction: Engaging Students in Learning
Common Core State Standards for Mathematical Practice
Students will:
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
The course satisfies these Mathematics, Science, and Technology standards:
Analysis, Inquiry, and Design
Mathematics
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Use Fraction Tiles to introduce and practice division of fractions
Develop the standard fraction division algorithm through discovery
Model how to apply efficient methods and strategies to find quotients by using mental math,
paper and pencil, or estimation and a calculator based on the context of the problem
Apply number sense to estimate answers
Teach students to use a fraction capable calculator to solve fraction problems after
estimating
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 21 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Addition and Subtraction of Decimals
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity.
Include strategies to meet diverse learning needs (differentiated instruction).
Video
Reflection prompts
Discussion forum
Quiz
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Use area models to visualize and add and subtract units and tenths, and units, tenths and
hundredths
Use linear models to visualize and add and subtract units and tenths, and units, tenths and
hundredths
Teach Trade First Algorithm to subtract units and tenths, and units, tenths and hundredths
Apply estimation skills to adding decimals
Teach students to choose appropriate methods to subtract decimals
Introduce games using the four-step model
Play the Units, Tenths and Hundredths Game
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 22 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Multiplication and Division of Decimals: Part 1
Paul Lawrence approaches multiplication and division of decimals in the same way he introduced
multiplication and division of whole numbers. He demonstrates linear and area models to increase
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
After completing this unit, educators will apply the following skills:
Teach multiplication and division of decimals using area and linear models
Model how to use number sense and estimation to determine a quotient
Use calculators with multiplication and division of decimals
Practice formats that require students to use number sense, operational knowledge of
decimals, and problem solving to determine
Teach students to choose efficient methods and strategies to solve mixed sets of operations
with whole numbers, fractions, and decimals by choosing from mental math, paper and
pencil, or estimation and a calculator
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 23 Date: self-paced Time: self-paced Number of hours for this session: 2 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Multiplication and Division of Decimals: Part 2
In this final unit, Paul Lawrence addresses two topics: how games can be used to teach math
concepts and how to use open-ended questions. He explains the use of open-ended questions and
evaluating written answers as a way to assess concept understanding and to prepare students to
succeed on constructivist standardized test questions. Lawrence also shares a nine-step program for
creating student-constructed responses. Finally, Lawrence reviews some of the overarching ideas
for the entire course and urges teachers to commit to using discovery-based math techniques in
their classrooms.
Objectives: Specify instructional goals and standards for each session.
Goals
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Resources (readings, artifacts, internet sites, videos, etc): Provide the title, author, edition, publisher,
cost, and where it is available. If there is a guest speaker, include the presenter’s name and affiliation.
Required Discovery-Based Math Manipulatives Kit ($69)
The supplementary kit includes a custom-tailored handbook to follow the online courses, as well as
correlating handouts, Discovery Templates, a Self-Study Guide, and the patented “Communicator,”
along with many manipulatives for hands-on activities. (The kit materials are packaged in a
convenient carrying case.)
Guest Speakers:
Calendar Session # 24 Date: self-paced Time: self-paced Number of hours for this session: 5 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date. Final
Take a unit of study for multiplying or dividing fractions, or addition, subtraction, multiplication or division
of decimals. Revise or replace it with discovery, inquiry-based, hands-on experiences that use manipulatives
of your choice to provide concrete and conceptual understanding before introducing symbols and the
traditional algorithm. Use the provided Lesson Thinking Template to plan your lessons. The Template can
be found in the Resources section of the e-Classroom.
Please include in your lesson:
1. Select a standard or learning objective for multiplying or dividing
fractions, or addition, subtraction, multiplication or division of decimals.
2. Outline the steps your lesson will follow that will move understanding from concrete, through iconic
to symbolic forms.
3. Identify the manipulatives, templates, games and game pieces, etc., that you will use. Adapt or use
the course templates and/or games if appropriate.
4. Include at least two activities (or games) to teach the concept in different ways to meet differentiated
learning needs. Identify the instructional need and describe the steps you will use to teach students
the activity or game.
5. Create 3 annotated sample practice pages for the lesson. Outline or script how you will introduce
each page and do a sample problem with students.
o Practice Page 1: 6-8 mental math problems. Annotate each problem. Note how you
expect students to describe mental math process to solve the problem.
o Practice Page 2: 6-8 problems to estimate the answers, each with 4 possible choices.
Annotate each problem. Note how you expect students to describe their estimation
process and how they selected the correct answer.
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201