A research-based Canadian professional learning initiative

1
Number and Operations
A research-based Canadian professional learning
initiative
Day 1 Slide 1
2
Three-Day Agenda
Day 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
3
Petals Around the Rose
4
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.

5
Roll 1

• The answer is 4.

6
Roll 2

• The answer is 6.

7
Roll 3

• The answer is 0.

8
Roll 4

• The answer is 4.

9
Roll 5

• The answer is 12.

10
Roll 6

• The answer is 0.

11
Want to keep going?

• http//illuminations.nctm.org/LessonDetail.aspx?id
  L576

12
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.

13
PRIME Map Research

• University-conducted research
• Canadian research and content
• Designed using current research on developmental
  learning and with KGrade 6 curriculums
  in mind
• Tested with thousands of KGrade 7 students in
  two stages of field testing
• KGrade 3 students were interviewed.
• Grades 47 students wrote written tests.

Day 1 Slide 4
14
Your Curriculum and Supporting Documents
Your Textbook/ Resources Implementation and
Support
PRIME

• Informs the teacher
• Focuses on meeting students needs in
  developmentally
• appropriate ways
• Provides support

Day 1 Slide 14
15
Benefits of PRIME
PRIME provides a framework to help you

• 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
16
Exploring the Kit
4 Poster Maps 2 for Number 2 for Operations
Guide to Using the Developmental Map Background
and Strategies book Diagnostic Tools booklet
Day 1 Slide 7
17
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
18
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
19
Number Map 5 Developmental Phases
PHASE 1 Beginner PHASE 2 Concrete PHASE 3 Whole Number Comfort PHASE 4 More Abstract PHASE 5 Flexible
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
20
What It Means To Be In a PhasePhase 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
21
Being in a phase describes a certain level of
mathematicalsophistication.
Day 1 Slide 12
22
Number Map8 Key Ideas
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
23
Number Map8 Key Ideas
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
24
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 writing
• multi-digit numerals (KS 3)

Day 1 Slide 15
25
The Number Map and Key Ideas

• PRIMEs 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
26
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
27
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?

Day 1 Slide 18
28
Lets take a health break.
29
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
30
What It Means To Be In a PhasePhase 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
31
How the Guide is Organized
Section 1 Introduction
Section 2 Phases of Development in
Number Section 3 Phases of Development in
Operations Appendices References
Day 1 Slide 21
32
How Sections 2 3 of the Guide are Organized
Each phase is organized by these headings

• Identifying Students in Phase X
• Supporting Students in Phase X
• Consolidating and Extending Phase X
  (Instructional Focus Suggestions)

Day 1 Slide 22
33
Linking Samples to Phases

• For each sample
• Locate the phase indicator(s) on the Number PI
  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
34
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.
Day 1 Slide 24
35
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
36
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 on
• each number line on your BLM.

Day 1 Slide 28
37
Race for Red, Version 1

• 1. 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
38
Race for Red, Version 2

• 1. 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
39
Race for Red, Version 3

• 1. 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
40
Lunch?
41
Self-ReflectionFractions

• 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
42
Introducing Fractions
Why fractions are important Fraction
contexts Manipulatives for teaching fractions
Day 1 Slide 33
43
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?

BS, p. 169
Day 1 Slide 34
44
Modelling Equivalence

• Use pattern blocks, a geoboard, or cubes.
• A. Show that is equivalent to .
• B. Show that is equivalent to .

Day 1 Slide 35
45
A Model for Tenths or Fifths
This represents a whole, or 1.
Day 1 Slide 36
46
A Model for
Day 1 Slide 37
47
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
48
Background and Strategies
Section 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
49
Fractions and Phases

• Use the Number PI Map. List the indicators that
  deal with fractions.

Day 1 Slide 40
50
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
Guide, pp. 32 and 33.
Day 1 Slide 41
51
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
52
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
53
Comparing the Number and Operations PI Maps

• Compare the two maps.
• How are they the same?
• How are they different?

Day 1 Slide 44
54
Operations Map 5 Developmental Phases
PHASE 1 Beginner PHASE 2 Concrete PHASE 3 Whole Number Comfort PHASE 4 More Abstract PHASE 5 Flexible
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
55
Operations Map6 Key Ideas
There are 3 key concepts Concept 1 Addition
leads to a total and subtraction indicates whats
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
56
Operations Map6 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.
Day 1 Slide 47
57
Relating Operations Through Key Concepts

• Key Concept 1 Addition leads to a total
• and subtraction indicates whats 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
58
Relating Operations Across the PhasesKey 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
59
Getting Ready for Day 2
Student Tasks Assign some of the tasks on the
next two slides to several students in your
class.
Day 1 Slide 50
60
Getting Ready for Day 2
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
61
Getting Ready for Day 2
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
all hit 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
62
Three-Day Agenda
Day 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