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
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.
There are 3 key skills:
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
Phase 1 Phase 2 Phase 3 Phase 4 Phase 5
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
Race for Red, Version 21. Roll the die and take that many
yellow counters.
2. Once you have 3 yellow counters, trade for 1 blue counter.
3. Once you have 3 blue counters, trade for 1 green counter.
4. Once you have 3 green counters, trade for 1 red counter.
The first one with 1 red counter wins.
Day 1 Slide 30
Race for Red, Version 31. Roll the die and take that many
yellow counters.
2. Once you have 10 yellow counters, trade for 1 blue counter.
3. Once you have 10 blue counters, trade for 1 green counter.
4. Once you have 10 green counters, trade for 1 red counter.
The first one with 1 red counter wins.
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.
Phase 1 Phase 2 Phase 3 Phase 4 Phase 5
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
There are 3 key skills:
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.
There are 3 key skills:
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
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.
Day 1 Slide 51
Getting Ready for Day 2Tasks for Junior 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 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
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 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
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