a research-based canadian professional learning initiative day 1 slide 1 number and operations

A research-based Canadian professional learning initiative

Day 1 Slide 1

Number and Operations

Three-Day AgendaDay 1

- Introduction- Overview of PRIME- Representing

a Number- The Number Maps- Key Ideas Across the

Phases- Fractions Across

the Phases- The Operations Map- Getting Ready for

Day 2

Day 3

- Introduction

- Diagnostic Tools from Day 2

- Fostering Communication

- Planning with PRIME

- Problem Solving with PRIME

- Reflecting Back/ Moving Forward

Day 2- Introduction

- Phasing Multiplication Tasks

- Student Tasks (Day 1)

- Algorithms Across the Phases

- Open & Choice Tasks

- Manipulatives Across the Phases

- The Diagnostic Tools

- Getting Ready for Day 3

Day 1 Slide 2

Petals Around the Rose

The Rules:•The name of the game is petals

around the rose.•The name of the game is

important.•The answer is either zero or even.•I can only repeat the rules or tell

you the answer.

Roll #1

The answer is 4.

Roll #2

The answer is 6.

Roll #3

The answer is 0.

Roll #4

The answer is 4.

Roll #5

The answer is 12.

Roll #6

The answer is 0.

Want to keep going?

http://illuminations.nctm.org/LessonDetail.aspx?id=L576

http://illuminations.nctm.org/LessonDetail.aspx?id=L576

Excerpt from Bill Gates’ original program:

PRINT "THE NAME OF THE GAME IS PEDAL AROUND THE ROSES"

No wonder he was having a tough time.

PRIME Map Research• University-conducted research

• Canadian research and content

• Designed using current research on developmental learning and with K–Grade 6 curriculums in mind

• Tested with thousands of K–Grade 7 students in two stages of field testing

• K–Grade 3 students were interviewed.

• Grades 4–7 students wrote written tests.

Day 1 Slide 4

Day 1 Slide 14

PRIME

Your Curriculum and Supporting

Documents

Your Textbook/Resources

Implementation and Support

- Informs the teacher- Focuses on meeting

students’ needs in developmentally -appropriate ways-Provides support

Benefits of PRIME

• understand the development of students’ mathematical thinking

• become more comfortable with the math you teach

• differentiate instruction to enable students to experience success more often

• see connections in order to focus on the “big picture” of mathematics

Day 1 Slide 6

PRIME provides a framework to help you

Exploring the Kit

Day 1 Slide 7

4 Poster Maps: 2 for Number

2 for Operations

Guide to Using the Developmental Map

Background and Strategies book

Diagnostic Tools booklet

Representing a Number

Represent each number as many ways as you can. Use pictures, words, numbers, and manipulatives.

5 1 0.1 321

Day 1 Slide 8

21

There are different, but equivalent, representations

for a number.

This is a key idea in mathematics. It is relevant for all number types and

across multiple grades.

Day 1 Slide 9

Number Map— 5 Developmental Phases

PHASE 1Beginner

PHASE 2Concrete

PHASE 3Whole

Number Comfort

PHASE 4More

Abstract

PHASE 5Flexible

Comfort with whole numbers to

10

Comfort with whole numbers to

100 modelled concretely

Comfort with whole

numbers to 1000 and

some fractions and

decimals

Comfort with whole

numbers greater

than 1000, fractions, decimals

Flexibility with

numbers

Day 1 Slide 10

What It Means To Be “In a Phase”

Phase 2 could look like this

Phase 1 Phase 2 Phase 3 Phase 4 Phase 5

C1

C2

C3

C4

C5

S1

S2

S3

Day 1 Slide 11

Being “in a phase” describes a certain level

of mathematicalsophistication.

Day 1 Slide 12

Number Map—8 Key IdeasThere are 5 key concepts:

Concept 1 Numbers tell how many or how much.

Concept 2 Classifying numbers provides information about the characteristics of the numbers.

Concept 3 There are different, but equivalent, representations for a number.

Concept 4 We use a number system based on patterns.

Concept 5 Benchmark numbers are useful for relating and estimating numbers.

There are 5 key concepts:

Concept 1 Numbers tell how many or how much.

Concept 2 Classifying numbers provides information about the characteristics of the numbers.

Concept 3 There are different, but equivalent, representations for a number.

Concept 4 We use a number system based on patterns.

Concept 5 Benchmark numbers are useful for relating and estimating numbers.

Day 1 Slide 13

Number Map—8 Key Ideas

There are 3 key skills:

Skill 1 Recites counting patterns.

Skill 2 Compares numbers.

Skill 3 Uses conventional symbols to describe numbers.


Skill 1 Recites counting patterns.

Skill 2 Compares numbers.

Skill 3 Uses conventional symbols to describe numbers.

Day 1 Slide 14

Key Concepts vs. Key Skills

Key skills are tools that relate to the application of key concepts. For example:

Compare the conceptual notion of the

patterns in the place value system (KC 4) with

the related skill of writingmulti-digit numerals (KS 3)

Day 1 Slide 15

The Number Map and Key Ideas

PRIME’s key ideas (key concepts and key skills)

• are meaningful “big ideas” that organize the content of the map in order to show underlying connections

• help teachers understand their curriculum

PRIME’s key ideas (key concepts and key skills)

• are meaningful “big ideas” that organize the content of the map in order to show underlying connections

• help teachers understand their curriculum

Day 1 Slide 16

Exploring Development

Look inside your envelope marked KC3. The cutouts inside the envelope fit across the row of the map marked on the envelope.

• Put the 5 cutouts in order across the map.

• Repeat with the envelope marked KS2.

Day 1 Slide 17

Examining the Number Visual Overview Map

Work in pairs.

Consider the following:

• What manipulatives are shown and where?

• What real-world contexts are used?

• What information does the map show?

• What else do you notice?

Work in pairs.

Consider the following:

• What manipulatives are shown and where?

• What real-world contexts are used?

• What information does the map show?

• What else do you notice?

Day 1 Slide 18

Let’s take a health break.

Phases and Indicators Developmental Map

Consider these questions as you look at the Phases and Indicators Map:

• How is it similar to the Visual Overview Map?

• What are some of the differences?

• Where/when might you use the Phases and Indicators Map? the Visual Overview Map?

Day 1 Slide 19

What It Means To Be “In a Phase”

Phase 4 could look like this


C1

C2

C3

C4

C5

S1

S2

S3

Day 1 Slide 20

How the Guide is Organized

Section 2: Phases of Development in Number

Section 3: Phases of Development in Operations

Appendices

References

Day 1 Slide 21

Section 1: Introduction

How Sections 2 & 3 of the Guide are Organized

• Identifying Students in Phase X

• Supporting Students in Phase X

• Consolidating and Extending Phase X (Instructional Focus Suggestions)

Day 1 Slide 22

Each phase is organized by these headings:

Linking Samples to Phases

For each sample:

• Locate the phase indicator(s) on the Number P&I Map and then in the appropriate indicator chart in the Guide.

• Discuss how the sample is representative of the indicator and phase.

Day 1 Slide 23

The Number Map and Key Ideas

The key ideas (key concepts and key skills) are meaningful “big ideas”

that organize the content of the Map to show underlying connections.

The key ideas (key concepts and key skills) are meaningful “big ideas”

that organize the content of the Map to show underlying connections.

Day 1 Slide 24

Key Concepts and Key Skills

• relate seemingly unrelated outcomes/expectations or math content/topics

• help teachers and students make important conceptual connections

Day 1 Slide 25

Where does it go?

Choose Task A or Task B.

Task A: Put the number 21 on each number line on your BLM.

Task B: Put the number 1.3 oneach number line on your BLM.

Day 1 Slide 28

Race for Red, Version 11. Roll the die and take that many

yellow counters.

2. Once you have 4 yellow counters, trade for 1 blue counter.

3. Once you have 3 blue counters, trade for 1 green counter.

4. Once you have 2 green counters, trade for 1 red counter.

The first one with 1 red counter wins.

Day 1 Slide 29


yellow counters.





Day 1 Slide 30


yellow counters.





Day 1 Slide 31

Lunch?

Self-Reflection—Fractions

Work on your own.

• What do you know about fractions? Record 5 to 10 things.

• Put your paper away until the end of this section, when you will revisit what you have recorded.

Day 1 Slide 32

Introducing Fractions

Why fractions are important

Fraction contexts

Manipulatives for teaching fractions

Day 1 Slide 33

Eating Candies by Fractions

Use manipulatives to solve this problem.

• Mark ate half of the candies in a bag.

• Leila ate of what was left.

• Now there are 11 candies in the bag.

• How many were in the bag at the start?

32

Day 1 Slide 34

B&S, p. 169

Modelling Equivalence

Use pattern blocks, a geoboard, or cubes.

A. Show that is equivalent to .

B. Show that is equivalent to .

63

21

51

102

Day 1 Slide 35

A Model for Tenths or Fifths

This represents a whole, or 1.

Day 1 Slide 36

A Model for 52

Day 1 Slide 37

The Background and Strategies Book

Support for fraction work:

Fraction Meanings p. 111

Fraction Principles pp. 104 to 106

Manipulatives pp. 114 to 116

Common Errors pp. 113 and 114

Day 1 Slide 38

Background and StrategiesSection 1 Introduction

Section 2 Instructional Issues

Section 3 Content Issues

Section 4 Developing Number Sense

Section 5 Problem Solving

Section 6 Communication

Section 7 Assessment and Evaluation

Section 8 Differentiating Instruction

Day 1 Slide 39

Fractions and Phases

Use the Number P&I Map. List the indicators that deal with fractions.

Day 1 Slide 40

Fractions and Phase 2

I-6: Names and interprets simple fractions to identify parts of a region modelled concretely and pictorially

I-11: Recognizes and creates concrete and pictorial models of some fractional equivalents

Day 1 Slide 41

Guide, pp. 32 and 33.

Fractions and Phase 2

• concrete and pictorial models

• fraction of a region meaning

• simple proper fractions

• equivalence of one half and two fourths

Day 1 Slide 42

Fractions Across the Phases

• Describe what a student understands about fractions in each phase.

• Record your responses on chart paper.

Include only a few points about each phase.Use pictures, symbols, numbers, and words.


Day 1 Slide 43

Comparing the Number and Operations P&I Maps

Compare the two maps.

• How are they the same?

• How are they different?

Day 1 Slide 44

Operations Map— 5 Developmental Phases

PHASE 1Beginner

PHASE 2Concrete

PHASE 3Whole

Number Comfort

PHASE 4More

Abstract

PHASE 5Flexible

Focus on counting to

solve problems

Formal operations

with numbers to

20; Concrete

operations with

numbers to 100

Formal operations with whole numbers; Concrete

operations with

decimals

Fluency with whole

number operations;

Formal operations

with decimals

Fluency with whole number & decimal

operations; Concrete

operations with

integers & fractions

Day 1 Slide 45

Operations Map—6 Key IdeasThere are 3 key concepts:

Concept 1 Addition leads to a total and subtraction indicates what’s missing. Addition and subtraction are intrinsically related.

Concept 2 Multiplication and division are extensions of addition and subtraction. Multiplication and division are intrinsically related.

Concept 3 There are many algorithms for performing a given operation with multi-digit numbers.

There are 3 key concepts:

Concept 1 Addition leads to a total and subtraction indicates what’s missing. Addition and subtraction are intrinsically related.

Concept 2 Multiplication and division are extensions of addition and subtraction. Multiplication and division are intrinsically related.

Concept 3 There are many algorithms for performing a given operation with multi-digit numbers.

Day 1 Slide 46

Operations Map—6 Key Ideas


Skill 1 Recalls facts.

Skill 2 Uses standard mental math and estimation procedures with multi-digit numbers.

Skill 3 Computes using pencil and paper with multi-digit whole numbers and decimals, without the aid of a calculator.


Skill 1 Recalls facts.

Skill 2 Uses standard mental math and estimation procedures with multi-digit numbers.

Skill 3 Computes using pencil and paper with multi-digit whole numbers and decimals, without the aid of a calculator.

Day 1 Slide 47

Relating Operations Through Key Concepts

Key Concept 1 Addition leads to a total and subtraction indicates what’s missing. Addition and subtraction are intrinsically related.

Key Concept 2 Multiplication and division are extensions of addition and subtraction. Multiplication and division are intrinsically related.

Day 1 Slide 48

Relating Operations Across the Phases—Key Concept 1

How do students relate addition and subtraction in Key Concept 1 in each phase?

Look for indicators in each phase that “indicate” a relationship between the operations.

Day 1 Slide 49

Getting Ready for Day 2

Student Tasks

Assign some of the tasks on the next two slides to several students in your class.

Student Tasks

Assign some of the tasks on the next two slides to several students in your class.

Day 1 Slide 50

Getting Ready for Day 2Tasks for Primary Students

1. Write a number used in math class today. Now write two numbers that are like it and two numbers that aren't. Tell why they're like it or not.

2. Write three different subtraction questions with an answer of 5. Make them as different as you can.

3. How many different ways can you make two towers with 12 cubes? Show all the ways using numbers, words, or pictures.

4. Use the digits 2, 3, and 4 to create as many different numbers as you can. Order them from least to greatest.

Tasks for Primary Students


2. Write three different subtraction questions with an answer of 5. Make them as different as you can.

3. How many different ways can you make two towers with 12 cubes? Show all the ways using numbers, words, or pictures.

4. Use the digits 2, 3, and 4 to create as many different numbers as you can. Order them from least to greatest.

Day 1 Slide 51

Getting Ready for Day 2Tasks for Junior Students


2. Write three different subtraction or division questions with an answer of 5. Make them as different as you can.

3. Leah throws exactly four darts. They allhit the target. a) What is the least score she can get? the greatest score?b) Find a minimum of three scores she could get that are between the least and greatest scores. Show your work.

4. Use the digits 3, 4, 7, and 2 to create as many different numbers as you can. Order them from least to greatest.

Tasks for Junior Students


2. Write three different subtraction or division questions with an answer of 5. Make them as different as you can.

3. Leah throws exactly four darts. They allhit the target. a) What is the least score she can get? the greatest score?b) Find a minimum of three scores she could get that are between the least and greatest scores. Show your work.

4. Use the digits 3, 4, 7, and 2 to create as many different numbers as you can. Order them from least to greatest.

Day 1 Slide 52

Three-Day AgendaDay 1

- Introduction- Overview of PRIME- Representing

a Number- The Number Maps- Key Ideas Across the

Phases- Fractions Across

the Phases- The Operations Map- Getting Ready for

Day 2

Day 3

- Introduction

- Diagnostic Tools from Day 2

- Fostering Communication

- Planning with PRIME

- Problem Solving with PRIME

- Reflecting Back/ Moving Forward

Day 2- Introduction

- Phasing Multiplication Tasks

- Student Tasks (Day 1)

- Algorithms Across the Phases

- Open & Choice Tasks

- Manipulatives Across the Phases

- The Diagnostic Tools

- Getting Ready for Day 3

Day 1 Slide 53