Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Number of hours for this session: 1 Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Setting Learning Goals
KDS has implemented a topic at the beginning of the course, Setting Learning Goals, that instructs participants
to read the syllabus that contains a course and unit descriptions and then to develop student learning goals related
to the domains and components from Charlotte Danielson’s Framework for Teaching (at
http://charlottedanielson.com/theframeteach.htm). They may also consult the NYC Teacher Effectiveness site (at
http://schools.nyc.gov/Academics/CommonCoreLibrary/Toolkit/TeacherEffectiveness/default.htm),
and the NYC CCSS (at
http://schools.nyc.gov/Academics/CommonCoreLibrary/Why/NYSStandards/default.htm). (Participants may also
draw from existing classroom data they have collected from pretests and assessments or know anecdotally.)
Objectives: Specify instructional goals and standards for each session.
Goal:
To set student learning goals
Method of Instruction: List the method of presenting: Classroom video or interactive hands-on activity. Include
strategies to meet diverse learning needs (differentiated instruction).
Participants consult standards
Participants create learning goals
Classroom Practice: Specify what skills and strategies the participant will bring back to his/her classroom.
Participants bring their learning goals to their students
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.
Charlotte Danielson’s Framework for Teaching (at http://charlottedanielson.com/theframeteach.htm).
NYC Teacher Effectiveness site (at
http://schools.nyc.gov/Academics/CommonCoreLibrary/Toolkit/TeacherEffectiveness/default.h
tm),
NYC CCSS (at
http://schools.nyc.gov/Academics/CommonCoreLibrary/Why/NYSStandards/default.htm). (Participants
may also draw from existing classroom data they have collected from pretests and
Guest Speakers:
Calendar Session # 2 Date: self-paced Time: self-paced Number of hours for this session: 2
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Classifying, Ordering and Exploring Real Numbers: Part 1
Paul Lawrence illustrates various ways that discovery-based instructional strategies can be
effectively applied to math instruction at the elementary level. In this presentation, Mr. Lawrence
considers the changes that have come about in math education in recent years: a move from rote,
procedure based, textbook learning to more constructivist, discovery based teaching and learning.
He suggests that math teachers need to adopt a more hands-on approach to instruction in which
students are presented with procedures only after they have mastered the concept that underlies the
procedure. In this presentation, he addresses the use of a variety of math manipulatives and
demonstrates how these can be effectively applied to helping students understand the base ten
number system.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this unit, educators will know:
Reasons for using math manipulatives
Uses of the Communicator™
Understand discovery-based methods to help students understand the base ten number
system
How to use discovery templates
How to help students understand how to order and compare sets of whole numbers
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
After completing this session, educators will know:
Why math manipulatives are essential for teaching math concepts
How discovery-based methods can substantially enrich math instruction
How geoboards and arrays can help teach essential math concepts
Why discovery-based approaches help students develop important higher order thinking
skills, e.g., problem solving and pattern seeking
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:
Use math manipulatives including geoboards and arrays to teach essential math concepts
Use discovery-based methods to enrich math instruction and help students develop
important higher order thinking skills, e.g., problem solving and pattern seeking
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)
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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 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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this session, educators will know:
How to introduce and help students visualize negative numbers
How to effectively teach scientific notation and its use in calculations without using the idea
of moving the decimal point
How to use investigation to teach order of operations
When to use PEMDAS in teaching the order of operations
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:
Use vertical timelines to introduce and compare positive and negative numbers
Teach scientific notation through discovery without using the procedure of moving the
decimal point
Explore order of operations through problem-solving investigations
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Use acronyms to summarize conclusions drawn
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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:
Use a variety of manipulatives to ground basic addition and subtraction facts in concrete and
iconic experiences
Develop understanding of addition and subtraction through the symbolic stage
Teach subtraction as counting on
Teach addition and subtraction using multiple strategies
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 # 7 Date: self-paced Time: self-paced Number of hours for this session: 2
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Topics: List session topic and material, e.g. handouts. Indicate midterm and final exam date.
Addition and Subtraction of Whole Numbers: Part 2
In this second unit on addition and subtraction of whole numbers, Paul Lawrence introduces several
games that can be used to help students master addition and subtraction with whole numbers. He
demonstrates how games build understanding of concepts and provide much needed practice in
ways that students enjoy. He also addresses the important number-sense skill of estimating and why
teachers should teach multiples strategies to solve mixed sets of multi-digit addition and subtraction
problems.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this unit, educators will know:
Why games are important teaching tools
The games “Win a Flat,” “Lose a Flat,” and “Column Addition”
The importance of teaching multiple strategies for solving math problems
How to help students develop their capacity to estimate answers to 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
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
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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 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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
How to move from the concrete to the iconic to the symbolic to teach math concepts
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:
Use concrete manipulatives to teach multiplication
Use Elinor Pinczes’ One Hundred Hungry Ants to develop math concepts
Develop students’ understanding of rows and columns
Teach multiple strategies for understanding math concepts
Teach strategies that move from concrete to iconic to symbolic
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
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
convenient carrying case.)
Guest Speakers:
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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:
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
In this third of three units on mastering multiplication and division facts, Paul Lawrence continues
to model how teachers can help students master the concept of multiplication and extends his
discussion to include division. He introduces several games and strategies to use with multi-digit
multiplication. Lawrence shows how students can develop the capacity to use mental math to solve
double-digit multiplication problems.
Objectives: Specify instructional goals and standards for each session.
Goals:
After completing this unit, educators will know:
Games that reinforce instruction in multiplication and division
Function machines that teach multiplication
How to teach multiplication using partial product algorithm
Steps to teach students to do mental math with multi-digit multiplication 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:
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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:
Use arrays to teach multi-digit multiplication
Teach the partial product algorithm to compute multi-digit multiplication
Employ strategies to estimate and validate estimations for multi-digit multiplication
problems
Have students use calculators appropriately
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:
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Please do the following:
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
Hands-on methods for teaching fractions and fraction equivalences
How to teach fractions using linear and area models
How to create unit sticks using construction paper and tape
Ways to use the unit stick to explore fraction equivalences and estimation
Classroom management techniques that support honest answers for estimating activities
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:
Use hands-on methods for teaching fractions and fraction equivalences
Teach fractions using linear and area models
Create fraction unit sticks with students using construction paper and tape
Apply the unit stick to teach fraction equivalences and estimation
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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 fractional relationships using rulers
Use rulers to illustrate fractional relationships and at the same time teach students to read a
ruler
Teach fraction equivalents using the Fraction Stick Template
Teach students to simplify fractions algebraically
Use calculators appropriately to simplify fractions
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:
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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:
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
same and compatible denominators. He demonstrates how concepts can be presented using Fraction
Tiles, rulers, and Fraction Stick equivalence charts. He connects the techniques he introduces to
more traditional approaches to instruction that use algorithms and he offers further guidance on
how to help students learn how to solve problems using mental math. Finally, he introduces several
games to reinforce the concepts covered in this unit.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
How to use Fraction Tiles, rulers, and equivalence charts to teach addition and subtraction
of fractions
How to transfer concepts from iconic experiences to mental math to solve addition and
subtraction problems involving fractions
Games to enhance and practice addition and subtraction of fractions with like and
compatible denominators
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
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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 multiplication of fractions using area models
Teach multiplication of fractions using Fractions Tiles
Use hand-drawn models to determine products of proper fractions
Guide students to discover the algorithm for multiplying fractions
Teach students to estimate products of fractions
Teach when and how to use calculators for multiplying fractions
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:
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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:
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
Paul Lawrence addresses addition and subtraction of decimals using the same sequence of
instruction to build understanding from iconic to symbolic without ever saying, “just line up the
decimals.” He models linear and area models to cement understanding of both addition and
subtraction of decimals. This methodology builds a foundation for algebra because students are
adding and subtracting like terms. Lawrence applies strategies from whole numbers, including
partial sums, trade first, same change, estimation, and games to teach decimals. He demonstrates
techniques and offers guidelines that can help students apply number sense to decide which method
is most appropriate to use in solving individual problems.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this unit, educators will know:
How to use area models to visualize and add and subtract units and tenths, and units, tenths,
and hundredths
How to use linear models to visualize and add and subtract units and tenths and units,
tenths, and hundredths
How to teach Trade First Algorithm to subtract units and tenths and units, tenths, and
hundredths
How to apply estimation skills to adding decimals
How to choose appropriate methods to subtract decimals
How to use a four-step model to teach games
How to play the Units, Tenths and Hundredths game
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
student understanding, without introducing the algorithm. He also models techniques to apply
number sense to estimating answers and explains why students should always be asked to estimate
before using a calculator. Lawrence offers guidelines that can help students decide which methods
are most appropriate to use in solving individual problems.
Objectives: Specify instructional goals and standards for each session.
Goals
After completing this session, educators will know:
How to teach multiplication and division of decimals using area and linear models
How and why number sense and estimation should be used to determine a quotient
When to 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 solutions
How to 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
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
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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 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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
After completing this unit, educators will know:
Games that will reinforce operations with decimals
How and why writing and open-ended questions are important in the math curriculum
How to develop student-constructed responses to open-ended questions
Key concepts for the course
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:
Play Decimal Roulette, Decimal Get a 100, and Dancing with Decimals games to reinforce
decimal math skills
Use open-ended questions in the math curriculum
How to assess student understanding by having them evaluate answers written by their peers
Reinforce key principles from the course
After School Professional Development Program 65 Court Street, Room 224, Brooklyn, NY 11201
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
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
Helaine Schwartz, Director [email protected]
Revised 11.2011 his
o Practice Page 3: Create a sample 6-8 problem You Decide with choices of the best way
to solve the problem using mental math (MM), paper and pencil (PP) or estimate and
calculator (EC). Include at least one of each type. Annotate each problem. Note how you
expect students to describe their thought process in choosing a method to solve the
problem.
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: