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Understanding Addition and Subtraction of Whole and Decimal Numbers Number Sense and Numeration, Grades 4 to 6 (with reference to Volumes 2 and 6) The Literacy and Numeracy Secretariat Professional Learning Series Session A – Activating Prior Knowledge 1. Aims of Numeracy Professional Learning 2. About Problem Solving 3. Learning Goals of the Module 4. Warm Up – What Ways Do We Use Math? 5. Activating Mathematical Knowledge – Problem #1 6. Reflect and Connect 2 Aims of Numeracy Professional Learning • Promote the belief that all students have learned some mathematics through their lived experiences in the world and that the math classroom is one where students bring that thinking to their work. • Build teachers’ expertise at setting classroom conditions where students can move from their informal math understandings to generalizations and formal representations of their mathematical thinking. • Assist educators working with teachers of students in the junior division to implement student-focused instructional methods to improve student achievement – as referenced in the Number Sense and Numeration, Grades 4 to 6. 3 Aims continued • Have teachers experience mathematical problem solving as a model of what effective math instruction entails by: – collectively solving problems relevant to students’ lives that reflect the expectations in the Ontario mathematics curriculum; – viewing and discussing the thinking and strategies in the solutions; – sorting and classifying the responses to a problem to provide a visual image of the range of experience and understanding of the mathematics; and – analysing the visual continuum of thinking to determine starting points for instruction. 4 Learning Goals of the Module During these session, participants will: • develop an understanding of the conceptual models of whole and decimal numbers; • explore, through problem solving, conceptual and algorithmic models of whole and decimal number addition and subtraction; • analyse and discuss the role of student-generated strategies and standard algorithms in the teaching of addition and subtraction with whole and decimal numbers; and • Identify, reflect, and connect strategies that contribute to an effective mathematics classroom. Familiar, Unfamiliar, Interesting 5 Number Sense and Numeration, Grades 4 to 6 Volume 1: The Big Ideas Volume 2: Addition and Subtraction Volume 6: Decimal Numbers Volume 3: Multiplication Volume 5: Fractions Volume 4: Division 6 In What Ways Does Number Sense and Numeration, Grades 4 to 6 Describe Addition and Subtraction? 1. List 2 ideas about addition and subtraction that are familiar. 2. List 2 ideas about addition and subtraction that are unfamiliar. 3. List 2 ideas about addition and subtraction that are puzzling. 4. Which curriculum expectations are focused on addition and subtraction of whole and decimal numbers? Book Walk 7 Warm Up – What Ways Do We Use Math? 1. Introduce yourself to anyone at your table you do not know. 2. Describe the different ways you have used addition and subtraction in your daily life over the past week. 3. Record one way per sticky note and be ready to share. 8 Warm Up continued 9 4. Sort and classify your group’s addition and subtraction examples. 5. Describe your sorting rule. Connecting informal, lived, embodied mathematics to formal mathematics Examples of Addition and Subtraction in Our Daily Lives label 1 label 2 label 3 label 4 concrete graph Activating Mathematical Knowledge – Problem #1 What Does Addition Look Like? Model using base ten blocks and an open number line. a) 13 + 18 b) 1.3 + 1.8 13 Show 2 different representations of the mathematical thinking as you evaluate each expression. Modelling and Representing 10 Activating Mathematical Knowledge – Problem #1 What Does Subtraction Look Like? Model using base ten blocks and an open number line. a) 97 − 58 b) 9.7 − 5.08 9.7 Show 2 different representation of solutions for Modelling and Representing each expression. 11 Reflect and Connect 1. Describe the strategies you used to solve addition problems. 2. Describe the strategies you used to solve subtraction problems. 3. a) How is adding whole numbers and decimal numbers similar? b) Different? 4. a) How is subtracting whole numbers and decimal numbers similar? Comparing Solutions b) Different? 12 Session B – Developing Conceptual Knowledge 1. Warm Up – A Knowledge Package for Addition and Subtraction 2. Developing Conceptual Knowledge – Problem #2 3. Organizing Different Solutions 4. Reflect and Connect 13 Warm Up – Knowledge Package for Addition and Subtraction What key understandings about whole and decimal numbers are needed to support learning about: a) decimal addition b) decimal subtraction c) the relationship between decimal addition and subtraction? Concept Web 14 Developing Conceptual Understanding – Problem #2 The veterinarian told Camilla that the mass of her puppy increased by 3.5 kg in the last month. If the puppy has a mass of 35.6 kg now, what was its mass a month ago? a) Solve this problem in 2 different ways. b) Show your work. Compare your solutions. How are they similar? different? 15 Reflect and Connect 1.What ways should the different solutions to this problem be organized for class discussion? a) Share your solutions. b) Sort and classify the solutions. c) Describe the sorting criteria you used. 16 Reflect and Connect continued 2. How does this problem help students develop a conceptual understanding of a) addition of whole and decimal numbers b) subtraction of whole and decimal numbers c) mathematical relationships between different solutions to a problem? 17 Session C – Making Sense of Alternative Algorithms 1. Warm Up – Defining an “Algorithm” 2. Alternative Algorithms – Problem #3 • Traditional Addition Algorithm • Partial-Sums Algorithm • Adding-Up Subtraction Algorithm • Left-to-Right Subtraction Algorithm 3. Reflect and Connect 18 Warm Up – Defining an “Algorithm” Algorithms: • are a structured series of procedures that can be used across problems regardless of the numbers; • promote accuracy; and • are efficient. Is this an addition algorithm? How do you know? 19 268 +483 6 14 11 7 4 11 7 5 1 Warm Up – Algorithms Decimal Number Addition •Is this addition algorithm different for decimals? •How do you know? 268 +483 6 14 11 7 4 11 7 5 1 20 2 6.8 + 4 8.3 6 14 11 7 4 11 7 5 .1 Alternative Algorithms – Problem #3 How and why do they work? Traditional Addition 1 1 1 1 348 +583 931 3.48 +5.83 9.31 Partial-Sums Addition 348 + 583 800 120 11 931 34.8 + 58.3 80.0 12.0 1.1 93.1 21 Alternative Algorithms How and why do they work? 22 Adding-Up Subtraction 724 – 379 7.24 379 + 1 – 3.79 3.79 + 0.01 380 + 300 3.80 + 3.00 680 + 40 6.80 + 0.40 720 + 4 7.20 + 0.04 345 3.45 Alternative Algorithms How and why do they work? Left-to-Right Subtraction 724 724 – 379 – 300 424 – 70 354 –4 350 –5 345 72.4 72.4 – 37.9 – 30.0 42.4 – 7.0 35.4 – 0.4 35.0 – 0.5 34.5 23 Apply These Strategies Joan says that the school will meet their fundraising goal if the hot lunch sales are within a range. Here are the numbers she is using. 1. Calculate using different algorithms: 439.56 + 88.1 Addition Algorithms a) traditional b) partial sums 439.56 – 88.1 Subtraction Algorithms a) adding up b) left to right 24 Reflect and Connect 25 • What might have students done to add? • What strategies would you use to promote understanding of addition? A. B. 1 8 4 439.56 439.56 + 88.10 417.66 + 88.10 411.110 C. 439.56 + 88.10 41117.66 Reflect and Connect continued • What might have students done to subtract? • What strategies would you use to promote understanding of subtraction? A B. 4 3 9.5 6 4 3 9.5 6 8 8.1 0 4 5 1. 4 6 – – 8 8.1 0 4 5 1.4 6 26 Session D – Using Mental Math Strategies 1. Warm Up – Sharing Strategies 2. Mental Math Strategies Made Explicit – Problem #4 Mental Math for Addition Mental Math for Subtraction 3. Apply These Strategies 4. Professional Learning Opportunities 27 Warm Up – Sharing Strategies 1. What strategies do you use to add and subtract whole numbers mentally? 257 + 39 257 − 39 2. What strategies do you use to add and subtract decimal numbers mentally? 25.7 + 3.9 25.7 – 3.9 28 Mental Math for Addition – Problem #4 How and why do they work? Adding On Compensation Constant Sum 136 + 143 236 + 297 153 + 598 136 + 100 = 236 236 + 40 = 276 276 + 3 = 279 236 + 300 = 536 536 – 3 = 533 Take 2 from 153 and add 2 to 598 151 + 600 = 751 So, 153 + 598 = 751 100 136 40 3 236 276 279 236 533 536 Strategies for Mental Addition 29 Mental Math for Subtraction – Problem #4 How and why do they work? Partial Subtraction 387 – 146 Compensation 387 – 100 = 287 287 – 40 = 247 247 – 6 = 241 547 – 300 = 247 247 + 4 = 251 So 547 – 296 = 251 6 40 241 247 547 – 296 100 287 387 247 251 547 Strategies for Mental Subtraction Constant Difference 598 – 153 Add 7 to 153 to make it 160. So, add 7 to 598. 605 – 160 = 445 Check: 153 + 445 = 598 30 Apply These Strategies Joan is wondering whether the class’ hot lunch sales were above or below the actual cost of the lunches, $637.45. Here are the numbers she is using. Calculate using different mental math strategies: 1. 637.45 + 219.18 2. 637.45 – 219.18 Addition: a) adding on b) compensation c) constant sum Subtraction a) partial subtraction b) compensation c) constant difference 31 Reflect and Connect 1. When and why would these mental addition strategies be useful? Not useful? Mental Addition a) adding on b) compensation c) constant sum 2. When and why would these mental subtraction strategies be useful? Not useful? Mental Subtraction a) partial subtraction b) compensation c) constant difference 32 Professional Learning Opportunities Collaborate with other teachers through: • Co-teaching • Coaching • Teacher inquiry/study View • Coaching Videos on Demand (www.curriculum.org) • Deborah Ball webcast (www.curriculum.org) • E-workshop (www.eworkshop.on.ca) 33