Course Syllabus Template All Courses = 36 Hours; minimum 6

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

http://www.kdsi.org/NYC

mailto:[email protected]



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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

http://charlottedanielson.com/theframeteach.htm

http://schools.nyc.gov/Academics/CommonCoreLibrary/Toolkit/TeacherEffectiveness/default.htm

http://schools.nyc.gov/Academics/CommonCoreLibrary/Why/NYSStandards/default.htm

http://charlottedanielson.com/theframeteach.htm



http://schools.nyc.gov/Academics/CommonCoreLibrary/Why/NYSStandards/default.htm



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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.


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



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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 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.



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.


Goals:



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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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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






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Guest Speakers:


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.


Goals:


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




Students will:






Attend to precision







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Mathematics



Video

Reflection prompts

Discussion forum

Quiz


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








Guest Speakers:


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.



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Goals:


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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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



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Use acronyms to summarize conclusions drawn








Guest Speakers:


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.


Goals:


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




Students will:




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Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:

Calendar Session # 7 Date: self-paced Time: self-paced Number of hours for this session: 2



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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.


Goals:


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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum



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Quiz



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








Guest Speakers:


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.


Goals:


Why they should use manipulatives to teach multiplication

Why it is important to use multiple strategies when teaching math concepts



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How to move from the concrete to the iconic to the symbolic to teach math concepts

Standards




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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









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Guest Speakers:



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.


Goals:


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




Students will:








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Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:





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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.


Goals:


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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz





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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








Guest Speakers:


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.


Goals:


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





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Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:



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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.


Goals


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




Students will:






Attend to precision






Mathematics



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Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:


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.



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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.







Guest Speakers:


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.



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Goals


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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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



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Use classroom management techniques to support estimating activities








Guest Speakers:



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.


Goals


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




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Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:



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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.


Goals


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




Students will:






Attend to precision





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Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:


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



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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.


Goals


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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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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








Guest Speakers:


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.


Goals


How to use Fraction Tiles and Fraction Sticks to teach addition and subtraction of fractions

with non-compatible and overlapping denominators



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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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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



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Guest Speakers:


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.


Goals


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






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Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:



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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.


Goals


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




Students will:






Attend to precision





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Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:


Addition and Subtraction of Decimals



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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.


Goals


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




Students will:






Attend to precision






Mathematics



Revised 11.2011 his



Video

Reflection prompts

Discussion forum

Quiz



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








Guest Speakers:


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



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.


Goals


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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts



Revised 11.2011 his

Discussion forum

Quiz



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








Guest Speakers:


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.


Goals



Revised 11.2011 his


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




Students will:






Attend to precision






Mathematics



Video

Reflection prompts

Discussion forum

Quiz



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



Revised 11.2011 his








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.



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.







Guest Speakers:

Course Syllabus Template All Courses = 36 Hours; minimum 6

Documents

Transcript of Course Syllabus Template All Courses = 36 Hours; minimum 6