Universal Journal of Educational Research 8(1): 105-111, 2020
DOI: 10.13189/ujer.2020.080112
http://www.hrpub.org
Using the "Identifying a Pattern" Strategy to Solve
Mathematical Word Problems of Proportional
Quantities at Grade 5 – Vietnam
Ngo Truc Phuong 1 , Nguyen Phu Loc 2,*
1
Faculty of Education, Bac Lieu University, Bac Lieu City, Vietnam
2
School of Education, CanTho University, Can Tho City, Vietnam
*
Corresponding Author: nploc@ctu.edu.vn
Received October16, 2019; Revised December 13, 2019; Accepted December 17, 2019
Copyright©2020 by authors, all rights reserved. Authors agree that this article remains permanently open access under
the terms of the Creative Commons Attribution License 4.0 International License
Abstract
Ratios and proportions that describe
relationships between quantities are the foundation for
students to understand and develop many concepts and
skills in mathemat ics. Therefore, they play an important
role in students’ learn ing Mathematics. This paper will
present the results of the comparison of mathematical word
problems relating to proportional quantities in
Mathematics textbooks of Vietnam and USA fro m the
perspective of pattern identification. It also indicates that
the design and implementation of some activities to teach
the word problems in wh ich the analysis to identify
patterns is a necessary strategy. The study results revealed
that using “identifying a pattern” problems is a useful tool
to promote problem - solving competency for elementary
school students in Vietnam.
Keywords Identifying a Pattern, Mathemat ical Word
Problem, Mathematical Pattern, Proportion, Proportional
Quantities, Solving Mathematical Word Problem
1. Introduction
In a study about students’ difficult ies in solving
mathematical word problems fro m their teachers’
perspectives, Seifi et al. (2012) showed that students in
elementary grades had a weak foundation because their
teachers taught them inappropriate strategies. In order to
overcome students’ difficult ies, most of the teachers
suggested helping students in teaching them to look for a
pattern. New Jersey Mathematics Curriculu m Framework
(1997) emphasizes that every Mathemat ics teacher needs
to assist students in recognizing, generalizing, and using
patterns that exist in numbers, in shapes, and in the world
around them.
It can be seen that identifying patterns in solving
problems is considered as a helpful strategy. Patterns often
appear in Primary Mathematics textbooks of America and
Singapore with a variety of act ivities, such as identifying
rules, extending patterns and solving word problems with
patterns. However, in Vietnamese primary textbooks, the
authors have not seen solving problems , where looking for
a pattern is a strategy.
According to the trend of international integration,
Vietnamese education is imp lementing a curriculu m,
which
transforms
content-based
instruction
to
competency-based education. Therefore, enhancing
problem solving competency for primary students becomes
an essential element in teaching and learning goals of
“General Education Program: Mathematics” in Vietnam.
In order to solve problems effect ively, elementary students
need to be provided with d iscovery tools, including the
strategy of looking for a pattern.
In this article, fro m analy zing using “identifying a
pattern” in solving word problems relating to p roportional
quantities in textbooks and implementing an experiment
with Grade 5 students, the authors believe that using
patterns could bring more fruitfu l for students in solving
mathematical word problems.
2. Literature Review
2.1. Problems of Identifying a Pattern
A mathematical pattern can be described as any
predictable regularity, usually involving nu merical, spatial
or logical relationships. In every pattern, the various
elements are organized in some regular fashion. Once a
pattern is established, it is easy to predict what happens
next because a pattern can be extended or continued after it
106
Using the "Identifying a Pattern" Strategy to Solve M athematical Word Problems of
Proportional Quantities at Grade 5 – Vietnam
has been identified (Mulligan, 2009).
According to Warren, “The power of mathematics lies in
relations and transformations which rise to patterns and
generalizations. Abstracting patterns are the basics of
structural knowledge and the goal of mathematics learn ing”
(cited by Mulligan, 2009). Because the patterns can be
found everywhere - in nature, nu mbers, and in shape, the
strategy of looking for a pattern is one of the most
frequently used strategies in solving mathematics
problems.
Pattern-based thinking, using patterns to analyze and
solve problems, is a powerful tool for doing mathematics in
primary schools level. It is suitable for describing
relationships and making the foundation for fu rther work
on with algebraic functions in higher grades. In grade 5-6
levels, the key components of pattern-based thinking are
exploring, analyzing, and generalizing patterns, viewing
rules and input/output situations as functions (New Jersey
Mathematics Curriculu m Framework, 1997, p.345). In
elementary schools, the strategy of looking fo r a pattern is
an extension of Drawing a Tab le and Creating an
Organized List. These are two strategies used in solving
mathematics problems.
Loc (2016) proposed the following model of identify ing
a pattern (see Figure 1).
Figure 1.
Model of identifying a pattern (Loc, 2016)
In the problem of “identifying a pattern”, students need
to find a rule fro m the connection of the data with the given
informat ion. Once the pattern has been identified, students
can predict what will happen next to find the solution.
To solve problems based on identifying a pattern,
students have two indispensable requirements, including (1)
determining quantities in the problem and the relationship
between them, (2) using visual representations, words,
symbols or numbers to describe the rule.
When students are faced with non-routine problems that
have no standard method to solve, they usually give up
easily because they do not know how to get started. The
ability to discover and analyze patterns becomes a critical
tool to help them be oriented. At this t ime, the problem of
“identifying a pattern” is a part of the process of solving
non-routine problems.
The process of solving a problem using the “identify ing
a pattern” strategy can be performed as follows:
List the given information and identify the required
information.
Make an organized list or create a table.
Determine the rule through analyzing data (the
process of “identifying a pattern”).
Use the pattern to find the missing information and
direct to the correct solution to the problem.
2.2. A Comparison between Textbooks of Vietnam and
America on Word Problems Relating to
Proportional Quantities from the Perspective of
Pattern Identification
In this part, the authors present results of the comparison
between Go Math Grade 5 & Grade 6 (A merican textbooks)
and Toan 5 (Vietnamese textbook) about problems of
directly proportional quantities (name in Vietnamese).
In
American
mathemat ics
curriculu m,
fro m
Kindergarten to h igh school, looking for a pattern is
considered one of the strategies that are needed to equip
students in solving mathematical problems in general and
word problems in part icular. The level of the crit ical
components of pattern-based thinking is identified to be
more and mo re co mp licated at the higher grades. Students
develop higher-level thinking throughout their work with
patterns in many types of problems. According to NCTM
(2000), in grades 5-6, students express understanding of
patterns, relations, and functions as follows: (1) Describe,
extend, and make generalizations about geometric and
numeric patterns; (2) Represent and analyze patterns and
functional relationships using words, tables, and graphs.
In Go Math Grade 5, the numerical pattern is part of the
standards of computational and algebraic thinking.
Identifying a pattern is one of the strategies used to solve
word problems of proportional quantities. The example is
illustrated in the figure below (see Figure 2).
Universal Journal of Educational Research 8(1): 105-111, 2020
Figure 2.
107
Problem 1 of identifying a pattern (Go Math Grade 5, p.560)
In the above problem, to assist students in finding
answers, a table o f nu merical data representing
relationships is proposed. In the table, there are 3 given
values and other values adding to make 3 regular
increasing sequences. There are two rules: (1) the number
of extra lives is increased by 3, and the number of gold
coins is increased by 6 after a level; (2) the number of gold
coins is twice as great as the number of extra lives in any
level. The rules are described by words and symbols.
It can be seen that the common way to identify rules in a
pattern according to Go Math Grade 5 is finding the
difference between two consecutive numbers and finding
out whether the numbers have been mult iplied or d ivided
by any given number. At the grade level, the concept of
proportions or ratios does not appear in describ ing patterns.
Go Math Grade 6 continues to solving this type of
problem with describing the pattern relating to equivalent
ratios (see Figure 3).
Figure 3. Problem 2 of identifying a pattern (GoMath Grade 6, p.235)
The problem asks to find the amount of gas used to travel
48 miles if using 2 gallons can travel 12 miles. In the table,
students have to find many numbers of a sequence whose
rule is hidden. The numbers in colu mns 2 and 3 are
intended to assist students in exploring the pattern faster.
By addition to 2, students can find the missing number of
gas used, but the answer is based on equivalent ratios
2
.
12 48
Go Math Grade 6 also presents another case of
equivalent ratios. That is finding a unit rate of t wo values.
When solving problems, students can use a unit rate to find
the unknown value. Finding a unit rate so metimes leads to
a decimal number, for example
30
30 : 20
1,5
1,5x24
20 24
20 : 20 24
1
24
1x24
24
(Go Math Grade 6, p.252)
In general, according to Go Math, the way to assist
students in describing patterns is using a data table to
represent terms of quantities. Fro m the given values,
students implement regular additions to consecutive terms
of each quantity by the same number. Based on this rule,
the relationship of mult iple or ratio between t wo quantities
is found out.
In order to solve the problem of proportional quantities,
while Go Math focuses on using a pattern, Toan 5 presents
two methods of verbally solving as follows:
Method of rate mentions a rate of t wo values of the
same quantity. For examp le, because 3 hours is three
times as much as 1 hour, if travelling 4 kilo meters is
in 1 hour then travelling 12 kilo meters is in 3 hours (in
this case, 3 is a factor).
Method of reducing to a unit aims to find the value of
1 unit. For examp le, if we travel 12 kilo meters in 3
hours then in 1 hour (1 unit of t ime) we travel 4
kilometres, so in 5 hours we travel 20 kilometres.
When the value of 1 unit of a certain quantity is decimal,
the method of reducing to the unit is not used because the
problem o f proportional quantities is learned earlier than
the concept of decimal nu mbers. Besides, if the value of the
quantity is not mu ltiple of other quantity, students do not
solve problems by the method of rate.
Toan 5 also illustrates two quantities that are directly
proportional in the following data table in Figure 4.
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Using the "Identifying a Pattern" Strategy to Solve M athematical Word Problems of
Proportional Quantities at Grade 5 – Vietnam
However, if the value of 1 unit is a decimal or there are no
divisible relationships between the given values, both two
these methods cannot be an option for students.
Furthermore, it is not always clearly visible to recognize
mu ltip le or div isible relationships between the given values
if they are large nu mbers. As a result, students make errors
in calculating, even give up solving problems.
3. The Research Questions
Figure 4.
p.18)
Illustration of two proportional quantities in Toan 5 (Toan 5,
In describing the relationship, the textbook does not
explain the principle of regular addition by the same
number to each term of quantity, nor does it mention the
ratio or mult iple between time and distance. The remark
orients students towards two methods of verbal solving
discussed above.
Through studying Go Math (Grades 5-6) and Toan 5, we
can see some differences in solving problems of
proportional quantities as follows (see Table 1):
Table 1. The differences in resolving problems between Go Math and
Toan 5
Use patterns in solving stage
A unit rate of quantity is always
found; it can be a decimal
number.
Find out the proportion of two
values in two quantities to
expand a pattern.
4. Methodology
4.1. Learning Activities
2.3. Comments
Go Math
Fro m the comments above, three research questions are
posed as follows
1) In Vietnam, for solving problems of p roportional
quantities, how can the strategy of using the problem
“identifying a pattern” be taught to fifth graders?
2) Are students interested in using the strategy of
“identifying a pattern”?
3) Is using the strategy of “identifying a pattern”
effective to proportional quantities problem?
T oan 5
Use words and operations in
solving stage
A unit rate of quantity is always
a whole number, not a decimal
number.
Find rate (multiple times) of
two values in the same quantity
It can be seen that GoMath mentions three necessary
aspects about proportions which Dougherty et al. (2017)
indicated as follows
(1) The relationship between the quantities in a ratio is
multiplication in nature (not addition);
(2) Although one of the numbers in the ratio is not a
factor (or mu lt iple) of the other number, a unit rate of
quantity is always found;
(3) Equivalent ratios are not necessary integral mu ltiples
of another ratio.
To answer the three research questions above, we
designed a study consisting of three learning act ivities in
order to teach solving word problems of direct ly
proportional quantities at Grade 5 by using the problem of
“identifying a pattern” (see Appendix)
4.2. Participants
Experimental subjects were 30 fifth graders in primary
schools in Bac Lieu city. These students have not still
learned the directly proportional quantities according to
Math 5 curriculu m. The study was carried out in October
2019.
4.3. Teaching Activities
In our study, there are three major activit ies carried out
in 2 periods (70 minutes).
Because Grade 5 students have not learned about
patterns, the first activity aims to introduce patterns,
describe their rule in words, and extend them. Students
approach the concept of the pattern via working out the
problems in three given tables. In table 1, students identify
a pattern relating to finding the relationship of map
distance to the real distance, wh ich was taught in Grade 4.
In Toan 5, although there is only one table illustrating Map distance and real distance have a directly proportional
the proportional relat ionship, it clearly describes a relationship. Table 2 is for t wo quantities that are not
numerical pattern. Most of the Vietnamese students are proportional, and students discover relat ionships between
taught to solve problems of proportional quantities by work t ime and the money earned (+1 in the terms of work
either the method of rate or the method of reducing to a unit. time, + 2 in the terms of money earned). Table 3 requires
Universal Journal of Educational Research 8(1): 105-111, 2020
students to recognize the proportional relationship between
the length and the width of the T-shirt. Another task in
Activity 1 is identifying the equivalent ratios of quantities
in each table for students to understand the ratio clearly.
Through these illustrations of patterns, it is expected that
the ratio of two d irectly proportional quantities is
recognized.
The second activity aims to instruct students in the
process of solving word problems by creating tables and
using the “identifying a pattern” strategy. The problem
designed has three tasks in relation to the proportional
relationship between the nu mber of books and the amount
spent. In tasks (1) and (2), students can use the method of
rate or the method of reducing to a unit to find answers. For
task (3), referring to the table with some given numbers
describing the rule in words, students find all the missing
numbers, and then find the answer to the task (2).
The third activity is for students to practice themselves
on solving two problems. In Problem 1, two known
methods are no longer effective; question b directs students
towards finding smaller values and the unit rate. Problem 2
is more d ifficu lt because students need to find out two
quantities, create a table, and d iscover relat ionships
between them to get the answer.
After Activity 3, the teacher asks students to propose the
general process used in solving word problems by using the
“identifying a pattern” strategy. Finally, students are
interviewed by asking the two following questions:
1) Do you like solving word p roblems with using the
strategy of “identifying a pattern”? Why or why not?
2) Are there any difficult ies you faced in solving word
problems with using the “identifying a pattern”
strategy?
4.4. Results
In Activity 1, students can quickly recognize rules and
fin ish finding missing numbers. For Table 1, students
discovered 3 types of pattern:
Answer 1: “The real d istance is always twice as long as
the map distance” (17/30 students).
Answer 2: “When the map distance is increased by 3, the
real distance is increased by 6” (10/30 students).
Answer 3: “The measure of map distance and real
distance is increased by the same factor" (3/30 students).
Similarly, in Table 2, all students describe the rule of
“the amount of earned money is increased by 2 after the
number of time work is increased by 1”. To find the
amount of earned money for 10 hours, 10/30 students
implemented mult iplying the amount of earned money for
5 hours by 2 because they base on “10 is a double of 5”.
Students’ wrong answers indicate that they have not
understood deeply about proportional quantities. The
others who found the amount of earned money for 6, 7, 8, 9
hours before 10 hours have a true answer.
Students describe the rule in Table 3 as fo llo ws: “the
109
length of each size is increased by 3 while the width is
increased by 2”. It is clearly shown that students did not
pay attention to the ratio of the length to the width.
In Activity 2, students prioritize the method of reducing
to a unit because they are familiar with this method in
solving problems in p revious grades. The first solving step
is finding the price of a book wh ich is 7000 VND, then
mu ltip lying the number of books by 7000 to find the
amount of money buying 24 books, buying 9 books. So me
students have difficu lty in finding the nu mber of books
corresponding to 126,000 VND due to the division of
126,000 by 7,000. There were no students who chose “24 is
twice 12” to find the amount of money for purchasing 24
books by multip lying the amount of money purchasing for
12 books by 2. It means that the rate of the numbers of
books is not chosen the first time. However, after working
with the table in task 3, students get the answer by finding
the amount of money for buying 9 books based on 3 books,
24 books based on 12 books. Although the rule in the table
described as “the amount of spent money is always equal to
the number of notebooks multip lied by 7000” or “the
number o f money is 7000 t imes the nu mber o f books”,
students found missing numbers based on the
mu ltip licat ion relationship between terms of the same
quantity.
In Problem 1 of Activ ity 3, based on “every 1500 people
will increase by 30 people”, students easily find out “every
500 people will increase by 10 people” and “every 50
people will increase by 1 person”. Fro m that, they fill in the
table of numbers according to the rule of increasing and
recognize “every 5000 people will increase by 100 people”.
Students also discovered that the ratio of increasing
population in the town is 1 out of 50.
In Problem 2 (Activity 3), students have difficu lty in
determining the variables and are confused about how to
present the solution. After creating the table, some students
find the answer but do not know how to present the solving
process.
To sum up, almost students perform requirements well in
Activities 1 and 2 like finding out the missing numbers and
describing patterns. However, there are still some errors in
words and meanings, such as “difference between the
number of notebooks is 7,000 VND” and “when the
amount of time work increases, the amount of money also
increases”. Students have some d ifficu lties in deciding
variables and presenting the solving process.
5. Discussions and Conclusions
After doing three learning activities with patterns,
students find out the general process to solve problems by
using pattern identification. It can be concluded that using
patterns helps students realize the constant ratio
relationship of two proportional quantities that the previous
methods do not. Moreover, extending patterns overcome
110
Using the "Identifying a Pattern" Strategy to Solve M athematical Word Problems of
Proportional Quantities at Grade 5 – Vietnam
some limitations on finding multiples as well as factors.
According to the results of the interview, 30/30 students
stated that solving problems by using the strategy of
“identifying a pattern” is more co mfortable than the
method of reducing to a unit and present fewer errors in
performing numerical operations because the numbers in
the table have clear rules. They agree that pattern
identification is an effective strategy for solving word
problems. However, students also state that it is difficult to
use words to describe patterns and present the solving
process in non-routine problems. According to the
common presentation in elementary school, each solution
for a problem must always include the solution sentences
and formula of numerical operations.
The positive results above indicate that Vietnamese
students should be approached using the method of
“identifying a pattern” in solving word problems as an
effective strategy. Besides, it is essential to introduce both
patterns of proportional and non-proportional quantities to
encourage students to identify many types of patterns.
In Vietnam primary mathemat ical syllabus, before grade
5, students are acquainted with the strategy of “identifying
a pattern” relating to numbers, shapes, forming tables of
addition and mult iplication, but this is not exploited in
solving word problems yet. Fro m the experimental and
survey results, we think that the teaching process designed
above is relatively appropriate.
In conclusion, to develop students' problem solving
competency through problem solving, it is very necessary
to provide them strategies and tools, including the method
of “identifying a pattern”.
2.
In each of the above tables, do you comment on the
ratio between the quantities in the given table?
Acti vi ty 2 : Sol ve Problems Using a Pattern Recogni tion
Known that if you buy 12 notebooks, it costs 84,000
VND.
1. How much does it cost to buy 24 notebooks?
2. How much does it cost to buy 9 notebooks? If you
have 126,000 VND, how many notebooks can you
buy?
3. Given the following table:
Number of
notebooks
Amount of
money (VND)
1.
Real distance (km)
6
12
…
27
…
18
*Table 2
Number of working hours
1
2
3
Earned money (in dollars)
4
6
8
4
5
10
12
84
thousand
24
252
thousand
Fill the given data in the following table:
Number of existing people
The number of people
increased by 1 year
*Table 1
6
6
Co mmune A has a population growth rate according
to the law: after 1 year for every 1,500 people, an
increase of 30 people. Ask if this co mmune has 5000
people, at this time next year how many more people
will increase?
a.
Describe the rule in the following tables and find the
missing numbers:
3
3
Fill the appropriate number in the blanks.
Describe the rule used to find the above numbers?
Use this table to find the answer to question 2?
b.
To choose some additional values for ro ws of
"existing people", which of the following
numbers would you choose?
50, 500, 1000, 3000, 4500, …
c.
Determine the rule for nu mber sequences in the
table and find the nu mber of people increasing
when commune A has 5000 people?
Activity 1: Familiarizing with the Patterns
Distance in the map (cm)
2
Activity 3: Solving Math Problems
Appendix
1.
1
2.
Solve the following problem by making a table:
“Students of Class 5/1 go to charity at Hematology
Hospital. They need to buy 200 bo xes of milk. Known that
in a supermarket, if buying 5 boxes of milk, they get 1 free.
What is the total number of milk boxes they need to buy?”
*Table 3. The length and width of Viet Tien men's T-shirts are as follows
Size
Length (cm)
Width (cm)
S
72
48
M
75
50
L
52
XL
54
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