Understanding Addition and Subtraction of Whole and Decimal Numbers

Understanding Addition and Subtraction of Whole and Decimal Numbers

The Literacy and Numeracy Secretariat Professional Learning Series

Number Sense and Numeration, Grades 4 to 6 (with reference to Volumes 2 and 6)

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Session A – Activating Prior Knowledge 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

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Aims of Numeracy Professional LearningAims 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.

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• 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.

Aims continuedAims continued

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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.

Learning Goals of the ModuleLearning Goals of the Module

Familiar, Unfamiliar, Interesting

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Volume 1: The Big Ideas

Volume 3: Multiplication

Volume 4: Division

Volume 5: Fractions

Volume 6: Decimal Numbers

Volume 2: Addition and Subtraction

Number Sense and Numeration, Grades 4 to 6Number Sense and Numeration, Grades 4 to 6

7In What Ways Does Number Sense and Numeration, Grades 4 to 6 Describe Addition and Subtraction?

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

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Warm Up – What Ways Do We Use Math? 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.

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4. Sort and classify your group’s addition and subtraction examples.

5. Describe your sorting rule.

label 1 label 2 label 3 label 4

Examples of Addition and Subtraction in Our Daily Lives

Connecting informal, lived, embodied mathematics to formalmathematics

Warm Up continuedWarm Up continued

concrete graph

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Model using base ten blocks and an open number line.

a) 13 + 18

b) 1.3 + 1.8

Show 2 different representations of the mathematical thinking as you evaluate each expression.

Activating Mathematical Knowledge – Problem #1

What Does Addition Look Like?

Modelling and Representing

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Model using base ten blocks and an open number line.

a) 97 − 58

b) 9.7 − 5.08

Show 2 different representation of solutions for each expression.

9.7

Activating Mathematical Knowledge – Problem #1

What Does Subtraction Look Like?

Modelling and Representing

12Reflect and ConnectReflect 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?

b) Different?Comparing Solutions

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Session B – Developing Conceptual Knowledge 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

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Warm Up – Knowledge Package for Addition and Subtraction 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

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Compare your solutions. How are they similar? different?

a)Solve this problem in 2 different ways.

b)Show your work.

Developing Conceptual Understanding – Problem #2Developing 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?

16Reflect and Connect 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.

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Reflect and Connect continuedReflect 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?

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Session C – Making Sense of Alternative Algorithms 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

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Warm Up – Defining an “Algorithm”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?

2 6 8+ 4 8 3

6 14 117 4 117 5 1

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• Is this addition algorithm different for decimals?

•How do you know?

2 6.8

+ 4 8.3

6 14 117 4 11

7 5 . 1

2 6 8+ 4 8 3

6 14 117 4 117 5 1

Warm Up – Algorithms Decimal Number AdditionWarm Up – Algorithms Decimal Number Addition

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Alternative Algorithms – Problem #3 How and why do they work?Alternative Algorithms – Problem #3 How and why do they work?

Traditional Addition Partial-Sums Addition

1 1

348

+583

931

34.8+ 58.3 80.0 12.0 1.1 93.1

348+ 583 800 120 11 931

1 1

3.48

+5.83

9.31

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Adding-Up Subtraction

7.24

– 3.79 3.79 + 0.01

3.80 + 3.00

6.80 + 0.40

7.20 + 0.04

3.45

724

– 379 379 + 1

380 + 300

680 + 40

720 + 4

345

Alternative Algorithms How and why do they work?Alternative Algorithms How and why do they work?

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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

Alternative Algorithms How and why do they work?Alternative Algorithms How and why do they work?

24Apply These StrategiesApply 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

439.56 – 88.1

Addition Algorithmsa) traditionalb) partial sums

Subtraction Algorithmsa) adding upb) left to right

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• What might have students done to add? • What strategies would you use to promote

understanding of addition?

439.56

+ 88.10

439.56

+ 88.10

439.56

+ 88.10417.66 411.110 41117.66

A. B. 1 8 4 C.

Reflect and ConnectReflect and Connect

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4 3 9.5 6

– 8 8.1 0

4 3 9.5 6

– 8 8.1 0 4 5 1. 4 6 4 5 1.4 6

A B.

• What might have students done to subtract? • What strategies would you use to promote

understanding of subtraction?

Reflect and Connect continuedReflect and Connect continued

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Session D – Using Mental Math Strategies 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

28Warm Up – Sharing Strategies Warm Up – Sharing Strategies

1. What strategies do you use to add and subtract whole numbers mentally?

2. What strategies do you use to add and subtract decimal numbers mentally?

257 + 39257 − 39

25.7 + 3.925.7 – 3.9

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Mental Math for Addition– Problem #4How and why do they work?

Mental Math for Addition– Problem #4How and why do they work?

Strategies for Mental Addition

Adding On

136 + 143

136 + 100 = 236236 + 40 = 276276 + 3 = 279

Compensation

236 + 297

236 + 300 = 536536 – 3 = 533

Constant Sum

153 + 598

Take 2 from 153 and add 2 to 598151 + 600 = 751So, 153 + 598 = 751

236 533 536136 236 276 279

100 40 3

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Partial Subtraction387 – 146

387 – 100 = 287287 – 40 = 247247 – 6 = 241

Compensation

547 – 296

547 – 300 = 247247 + 4 = 251So547 – 296 = 251

Constant Difference598 – 153

Add 7 to 153 to make it 160. So, add 7 to 598.605 – 160 = 445Check: 153 + 445 = 598

247 251 547241 247 287 387

6 40 100

Mental Math for Subtraction– Problem #4How and why do they work?

Mental Math for Subtraction– Problem #4How and why do they work?

Strategies for Mental Subtraction

31Apply These StrategiesApply 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 onb) compensationc) constant sum

Subtractiona) partial subtractionb) compensationc) constant difference

32Reflect and ConnectReflect and Connect1. When and why would

these mental addition strategies be useful? Not useful?

2. When and why would these mental subtraction strategies be useful? Not useful?

Mental Addition

a) adding on

b) compensation

c) constant sum

Mental Subtraction

a)partial subtraction

b)compensation

c) constant difference

33Professional Learning OpportunitiesProfessional Learning OpportunitiesCollaborate 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)