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TeachingMasteryto
MatheMatics
Yeap Ban Har, PhD
Seashells ProblemJuan has 3 seashells more than Kim. Juan and Kim have 15 seashells altogether. Find the number of seashells that Juan has.
2 units = 15 – 3 2 units = 12 1 unit = 12 ÷ 2 = 6Kim has 6 seashells. So, Juan has 9 seashells.
3Juan’s seashells
Kim’s seashells 15
Bar Modeling From Research to PracticeAn Effective Singapore Math Strategy
A Problem-solving Tool
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The first in the series of Mathematics professional development titles, Bar Modeling: A Problem-solving Tool is written for educators who want to know the ‘How’ and the
‘Why’ of bar modeling — a visual problem-solving heuristic which serves as a foundation to
algebraic thinking — as well as the pedagogy of the heuristic.
With its inductive approach to helping readers discover and understand the use of bar models,
this book serves both as a guide and as a resource for practice in the use of the model method
in solving word problems.
In the first introductory chapter, readers get a brief overview of bar modeling, more commonly
known as the model method. In subsequent chapters, users will get to explore and work on
the different types of bar models, starting from the basic part-whole model form to more
challenging problems involving various types of models in one question. Each new concept is
introduced via numerous Examples, followed by Guided Practices and Independent Practices.
Notes in the sidebars as well as tips and hints provide reinforcement as well as guidance while
attempting the word problems.
Readers will also be able to get an insight into the theoretical underpinnings of the model
method based on research conducted by educators in this field, as well as learn how the use
of the model method can be brought into the classroom in various grade levels.
Preface
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ContentsChapter 1: The Model Method 1
The Model Method: An Introduction 2The Model Method and Arithmetic Word Problems 4The Model Method and Algebraic Word Problems 5Learning From Research 7Research Study: The Model Method in Singapore Schools 8
Chapter 2: Part-Whole Model 9Part-Whole Model: An Introduction 10Part-Whole Models Involving Discrete Quantities 11Learning In The Field 14Part-Whole Models Involving Continuous Quantities 15Challenging Part-Whole Situations 22Learning From Theory 28Solutions 29
Chapter 3: Comparison Models 32Comparison Models: An Introduction 33Additive Comparison Model 36Multiplicative Comparison Model 51Additive and Multiplicative Comparison Models 65Learning In The Field 69Task List 70Comparison Models Involving Fractions, Ratio, and Percent 74Learning From Theory 85Learning From Research 87Solutions 90
Chapter 4: Before-After Model 97Basic Before-After Models 98Before-After Models Involving Fractions 104Before-After Models Involving Percent 112Before-After Models Involving More Than One Quantity 116Learning From Theory 123Learning From Research 124Learning In The Field 126Solutions 128
Chapter 5: Advanced Skills In Model Method 130Shifting the Bar 131Cutting the Bar 132Cutting and Shifting the Bar 147Putting It All Together 154Learning From Research 158Learning From Theory 162Learning In The Field 163Solutions 165
References 167
Word Problem Index 168
Keyword Index 169
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Chapter 1The Model Method
This chapter introduces the model method (bar modeling) as a tool to help students solve arithmetic and algebraic word problems. The model method is a common problem-solving heuristic used in primary schools in Singapore.
Synopsis The Model Method:
An Introduction
The Model Method and Arithmetic Word Problems
The Model Method and Algebraic Word Problems
“The Model Method for problem solving... was an
innovation in the teaching and learning of mathematics
developed by the [Curriculum Development Institute of
Singapore, Ministry of Education] project team in the 1980s
to address the issue of students having great difficulty
with word problems...”(Kho, Yeo, & Lim, 2009, p. 2)
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The Model Method: An Introduction
In the early grades, students can solve word problems such
as the Cupcakes Problem below, by acting out the situation.
CupcakesProblem
has 2 cupcakes. has 3 cupcakes.
How many cupcakes do and
have altogether?
Initially, cupcakes are used to
illustrate the problem.
Subsequently, generic concrete
materials such as connecting cubes
are used to represent the cupcakes.
Later, pictorial representations of
the number of cupcakes are used.
The pictorial representations become
increasingly less realistic and more
abstract.
2
3
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The model method was introduced in primary-level
mathematics textbooks in Singapore in the early 1980s.
Today, it continues to be a major part of mathematics
education in the country.
Using rectangular bars to represent known or unknown
quantities, the model method is a common diagrammatic
problem-solving tool.
In this book, the rectangular bars in the models are drawn
proportionally to one another. In practice, there should
not be an over-emphasis on drawing the bars to the exact
proportion but estimate to the best of the students’ ability.
In the following sections, we will see how bar models can be
used to solve both arithmetic and algebraic problems.
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The Model Method and Arithmetic Word Problems
✎Let us revisit the Cupcakes Problem (p. 2).
CupcakesProblem
has 2 cupcakes. has 3 cupcakes.
How many cupcakes do and
have altogether?
Solution
?
2
3
2 + 3 = 5
and have 5 cupcakes altogether.
The solution shown above includes the use of the model
method. In arithmetic word problems such as this one,
the model method helps students visualize the situations
involved so that they are able to construct relevant number
sentences.
Apart from problem comprehension, the model method is
also a form of problem representation that helps students
gain a deeper understanding of the operations they may use
to solve problems.
Instead of relying on keywords and superficial features
of a word problem, the model method helps students
see the relationships between and among the variables
in the problem.
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The Model Method and Algebraic Word Problems
✎The model method lays the foundation for learning formal algebra. In the Age Problem below, rectangular bars are used to represent an unknown quantity.
AgeProblem
Jake is 3 years older than Kyla and 2 years younger than Larry. The total of their ages is 41 years. Find Jake’s age.
Solution
?
Jake’s age
3
Kyla’s age
2
Larry’s age
41
3 units = 41 + 3 – 2 = 42
1 unit = 14
Jake is 14 years old.
In using the model method with algebraic word problems, we
can see three distinct advantages:
It helps students derive algebraic expressions.
It helps students construct algebraic equations.
It helps students simplify algebraic equations.
Jake is 3 years older than Kyla,
so the bar that represents Kyla’s
age is shorter than the bar that
represents Jake’s age.
Jake is 2 years younger than
Larry, so the bar that represents
Larry’s age is longer than the bar
that represents Jake’s age.
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