A research-based Canadian professional learning initiative Day 1 Slide 1 Number and Operations

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

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?

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 Day 1 Slide 8

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 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

What It Means To Be “In a Phase” Phase 2 could look like this Phase 1Phase 2Phase 3Phase 4Phase 5 C1 C2 C3 C4 C5 S1 S2 S3 Day 1 Slide 11

Being “in a phase” describes a certain level of mathematical sophistication. Day 1 Slide 12

Number Map—8 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

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. 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 writing multi-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 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? 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 1Phase 2Phase 3Phase 4Phase 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. 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 on each number line on your BLM. Day 1 Slide 28

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

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

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

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? 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. Day 1 Slide 35

A Model for Tenths or Fifths This represents a whole, or 1. Day 1 Slide 36

A Model for Day 1 Slide 37

The Background and Strategies Book Support for fraction work: Fraction Meanings p. 111 Fraction Principlespp. 104 to 106 Manipulativespp. 114 to 116 Common Errorspp. 113 and 114 Day 1 Slide 38

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

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 1Phase 2Phase 3Phase 4Phase 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 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

Operations Map—6 Key Ideas 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. 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. Day 1 Slide 50

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

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

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