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Transcript
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