A solid understanding of equivalent fractions is considered a
steppingstone towards a better understanding of operations with
fractions. In this article, 55 rural Australian students'
conceptions of equivalent fractions are presented. Data collected
included students' responses to a short written test and follow-up
interviews with three students from each year. This exploratory study
found most participating Years 4, 6 and 8 students were familiar with
geometric area models, particularly circles, and able to explain
equivalent fractions when presented geometrically as area models but had
difficulties when equivalents were presented numerically as a/b.
Many studies found middle primary and junior secondary students
have difficulties understanding, and working with, fractions. According
to Niemi (1996, p. 6), fractions, because of their importance, are
conventionally introduced to children in kindergarten and continue to
occupy a prominent place in school curricula from the second year of
primary. The concept of fraction is very important in understanding
equivalent fractions. Although learning equivalent fractions is repeated
in subsequent years, Kamii (1994, p. 2) found the performances of middle
primary years and junior secondary students are still disappointing.
This paper reports how some Years 4, 6 and 8 students view some fraction
models and simple equivalent fractions.
Three classes, from an Independent school in regional Australia who
agreed to participate in the study, were selected to represent the
learning continuum from the formal introduction of simple fractions in
Stage 2 (Years 3-4) through to operations with more complex fractions in
Stage 4 (Years 7-8) (NSWBOS, 2002). Fifty-five students (21 Year 4, 12
Year 6, and 22 Year 8), who had signed consent forms, undertook the same
paper-and-pencil test. Three students, each recommended by their
teachers and representing different ability levels--high, medium and
low--were selected for individual interviews following the test. The
test (Kerslake, 1986), comprised six questions on fraction models and
equivalent fractions. All students answered the same test and
semi-structured interview questions. The latter (Kerslake 1986) built on
students' test responses with four additional questions presented
as placards to further explore student understanding. Tests were
administered during their 20-minute mathematics periods. Individual
interviews, conducted in their teachers' presence, were audio
This paper presents students' responses to questions that
focused on a general understanding of fractions and simple equivalent
fractions. Data collected also provided information on students'
conceptions of other, more complex, equivalent fractions and addition of
simple fractions. This is not discussed further in this article.
Question 1: Choose and tick the correct ways of saying fraction 2/5
from the following sentences.
(a) Two fifths, (b) Two over five, (c) Two by five, (d) Two upon
five, (e) None of the above.
Students were most familiar (Table 1) with "two fifths"
though three Year 4 students omitted it. Description "two over
five" was quite popular in Years 6 and 8 but less popular in Year
4. Some Year 6 and a few Year 8 students chose "two by five."
Everyone who chose "two fifths" explained it was the way
they were taught:
"[A]s if I was looking at a pizza cut into five and two fifths
have pepperoni. That is the way I have always been told to say it."
"[B]ecause I was taught that way and also I read the top
number first and the bottom number next." (Year 4)
Those who accepted "two over five" explained:
"Two is on a line above 5 so you say 2 over 5 or two
fifths." (Year 8)
"There is a two over a five" (Year 6)
A Year 4 student said, "The teacher taught us that two over
five is two fifths."
Overall, "two fifths" was popular with the majority of
Years 6 and 8 students also choosing "two over five." Students
seemed to reproduce the sanctioned, classroom language confirming the
(a) pedagogical influence on students' developing fraction
language, and (b) different capacities of students to explain their
understanding depending on their year level.
Question 2: Which of these pictures would help you know what the
fraction 2/5 is?
Models A and C, in that order (Table 2), were accepted by most
students in each year. Least selected by Year 4 is model D, while Years
6 and 8 students were certain it was incorrect.
Increased percentages, especially from Years 4 to 6 for C and B,
and highly consolidated A (1) reflect the pedagogical mediation of
student learning along a developmental continuum (from exploration and
introduction to consolidation) of types of:
1. geometric area models (with increased exposure to, and
consolidation of, rectangle-models to match established circle-model
2. fraction models (increased exposure to discrete models (i.e.,
part of a set or collection) to match established area-model
However these trends gradually decreased (B and C) from Years 6 to
8 though percentages for A and C were still relatively high with just
over half of Year 8 students selecting B. Students' conceptions,
and by implication the teaching of fraction models, peak at Year 6 with
a tapering off post-Year 6.
Example justifications (Figure 1) show that the first Year 8
student confidently reasoned that the unfamiliar B was also correct.
This is in contrast to the Year 6 student's certainty that seemed
to reflect a familiarity with the three different representations.
Interestingly, the second Year 8 student (Figure 1) excluded B while the
second Year 4 student chose only A. Students' reasoning, as
inferred from these explanations, reflect the implied pedagogical trends
Two Year 4 students who incorrectly chose D, said, "D helps me
to know two over five because there are 2 balls over five," and,
"'I think all of them help me, especially D because it has got
5 bottom and 2 above." Conceptually, these students interpreted 2/5
as representing two unrelated whole numbers, not as a part-whole
relationship, and appeared not to have developed any deeper
understanding of pictorial and numerical representations beyond their
visual spatial features.
Question 3. What fraction is shaded in each diagram given in A and
B? Are they equal? Explain.
Most students wrote correct fractions for all diagrams as
representing 1/2, by considering the partitioning of the regions into
equal parts. For example, two Year 8 students explained: "because,
in both, 2 parts out of 4 are shaded, indicating 2/4 . When simplified,
2/4 can be expressed as 1/2" and "because there are 2 shaded
shapes in each circle". In comparison, a Year 6 pointed out they
are equal "because both have half of them shaded" while a Year
4 reasoned "because there are 2 parts shaded in each picture and
they both equal a half." Overall, students found establishing
equality of halves easy.
Question 4. a. Draw a model to represent 2/3 and 4/6.
b. What could you tell by comparing your models?
Almost all students (Figure 2) correctly represented each fraction
with area models such as circles (most common) and rectangles,
suggesting that representing thirds and sixths diagrammatically was
easy. Describing equivalence between pictorial models was progressively
easier, as expected, from Years 4 to Year 8 (e.g., 76.2% Year 4, 33.3%
Year 6 and 27.3% Year 8 students could not describe equivalence).
According to the K-10 NSW Mathematics Syllabus, at Stage 3 (Years
5-6), children learn modelling, comparing and representing the new
fractions (thirds, sixths and twelfths) and finding equivalence using
pictorial representations (NSWBOS, 2002). Percentages suggest the
development of students' conceptions (fraction models and
equivalence), seem to occur more steeply between Years 4-6 than between
Years 6-8. Some responses (Figure 2) indicated (a) the dual use of
"amount" to describe both area and number of parts to justify
equivalent fractions and equal areas (circle, Year 8 students), and (b)
Year 4 students' (circle, first student; rectangle student)
contextually bound knowledge and difficulty abstracting meaningful
relationships. The decline in percentages (Years 4-8) is expected, given
curricular expectations for Stage 2 (Years 3-4), namely, thirds and
sixths are yet to be introduced.
[FIGURE 2 OMITTED]
Question 5. Are the following shaded circles equal? Compare and
explain your answer?
Every Year 6 and Year 8 student (Table 3) could represent the
shaded parts as fractions but a few incorrectly explained equivalence.
Some justified equivalence by simplifying fractions (procedural) and
some by matching areas (visual spatial). Although Year 4 students could
represent fractions, 71.4% gave incorrect explanations. Example
explanations (Table 4) and trends (Table 3) reflect students'
levels of understanding, along a developmental continuum of learning
fractions, across the primary-early-secondary years. Pedagogically,
teachers develop and consolidate student understanding of equivalence
and extend fractions to include halves, quarters and eighths in Stage 2
(Years 3 and 4) through modelling and pictorial representations, with
expansion to thirds, sixths and twelfths in Stage 3 (Years 5 and 6),
whilst increasingly more sophisticated justifications are expected from
Stage 4 (Years 7 and 8) students (NSWBOS, 2002).
Overall, some students have correct visual spatial representations
of equal areas, and therefore of equivalent fractions (column 2, Table
4) In contrast, incorrect explanations indicate that some students view
fractions additively as two unrelated whole numbers, where more pieces
means the two areas, and therefore fractions are different (column 3,
Table 4). The correct explanations (column 2, Table 4), in contrast,
indicate some (Years 6 and 8) students could justify equivalence, not
only geometrically, but also procedurally through simplification and
division. The number and sophistication of interpretations progressively
increase from Years 4 to 8.
Question 1. How would you explain to your friends what a fraction
Responses varied with most referring to "parts of a
whole," while one mentioned "part of a number" and
another "one number over another" (Table 5, each letter (A, D,
E, F, G, H and J) represents a student).
Year 4 students related "whole" to examples such as
pizza, circles and orange as evident below:
"OK, well, a fraction is basically two thirds or say you had a
piece of like a circle things and well we have a whole and we can cut it
into half to make it two and that would be two, uh... You cut it into
four that would be four, uh ..." (E)
"Uh ... a fraction is somewhat like you have an orange: you
cut them into half and you could cut into certain amount of pieces and
however many pieces out of one that would be say you cut them into six
pieces, that would be um ..." (F)
For Year 8, J described a fraction as "one number on another
... a part of whole," while H said, "a part of a number, a
part of a ... like a section part."
Students (J, A, D, E and F) predominantly described fractions as
"part of a whole," experientially by Year 4 (E and F) while
Year 8 students used other descriptors such as "part of a
number" (H), suggesting a connection between fractions and numbers,
and "one number over another" (J), which acknowledges the
numerical notation a/b. One Year 8 student could not define fractions.
Question 2. Have you come across this picture somewhere? What does
it tell you?
Years 4 and 6 student remembered seeing the picture either in
textbooks or on classroom walls. Most students did not discuss
equivalent fractions unless prompted. Instead, their observations were
limited to stating halves, thirds and sixths. One Year 6 student (A)
explained equivalent fractions when looking at the chart: "Oh it
shows equivalent fractions, so like one half, like one out of two is
equal to two out of four and eight out of ... Oh, four out of eight and
six out twelve. So that is exactly the same."
No Year 8 student remembered seeing the chart, suggesting they had
not used it recently. For example, "Um ... the numbers ... they are
um ... I don't know really ..." (H). The wall chart shows
linear models of fractions as lengths (i.e., models are line segments),
or arguably, a stack of rectangle area models (models are rectangles).
Test Question 2 revisited: Which of these pictures would help you
know what the fraction 2/5 is?
Responses indicated misconceptions particularly with D. They all
stated it was two out of five because of the two shaded circles on top
of the five unshaded circles. For example:
"Um ... I found it harder to use because two shaded ones are
at the top not in like five ..." (J, Year 8)
"That one's more than five because it's two on top
of five and that makes more sense." (F, Year 4)
"Because, uh ... on this one it is five there and on this one
it's got two and then five down the bottom. It's easier for me
to say that there is five there with this two shaded, so it is two
shaded out of five here." (E, Year 4)
E's explanation, although a misconception, suggests another
viable description for 2/5, namely, "two out of five"; one
that descriptively emphasises the part-whole relationship. Some students
may have been distracted by the oval and box around the discrete models
(B and D). However, this appeared unlikely in this survey as no student
raised it as a concern, either through the test or in the interviews.
Partial results from this study showed geometric area models,
representing "part of a whole" were the most familiar to the
participating students, particularly circles and, to a lesser extent,
rectangles. The discrete model was less common while most (of those
interviewed) had not recently used or seen a fraction wall chart.
Kerslake (1986) and Cramer and Henry (2002) also found children
predominantly selected circle models over discrete and linear models in
Responses showed some students perceive the numerator and
denominator as two separate, unrelated whole numbers, which subsequently
led to misconceptions when comparing area and numerical representations
of equivalent fractions.
Results reported here showed students found it easy to accept
equivalent fractions (halves, thirds and sixths) when presented
geometrically (test questions 3 and 5), but misconceptions emerged when
comparing thirds and sixths numerically (2/3 and 4/6, test question 4a)
and explaining equivalence (test questions 4b and 5). For example,
students reasoned circles divided into thirds and sixths are the
"same" but some felt that, numerically, "4/6 is double
2/3", suggesting the student probably meant "double" the
number of parts but leaves unstated the size of the part (i.e., does the
student mean: 4/6 = 2(2/3), 4/6 = 2/2 (2/3), or 2(1/6) = 1/3 ?) and,
"they are not equal they are added (doubled) on," indicating
misconceptions of fractions as representing double counts of unrelated
numbers and also their transitional understanding between developmental
stages of the continuum of learning fractions. These misconceptions
suggest that, pedagogically and conceptually, there is a need to develop
and consolidate students' understanding of (i) fractions, using
multiple models, as representing a relationship between the numerator
and the denominator and in addition, when comparing fractions, and (ii)
the relationship between number of parts, and size of parts.
These findings support those by Kerslake (1986), namely, students
recognised instances of equivalent fractions when presented in geometric
form, however, there was some conflict between the awareness that, for
example, 2/3 and 10/15 were the "same," and the feeling that
10/15 was bigger than 2/3 because 10/15 was 5 times bigger than 2/3.
Larson (1980) also indicated that, for seventh graders 2/6 was not seen
as having the same meaning as 1/3.
The results also reflected the timing of the implemented curriculum
on fractions (NSWBOS, 2002). For example, an increasing proportion of
students from Years 4 to 8 correctly explained equivalence between
circle-models of 2/3 and 4/6 with students showing the least percentage
in Year 4, the level where the equivalent fraction concept is formally
introduced, and the highest at Year 8. Also the lesser (B) and least
common (fraction wall) models indicate classroom pedagogy across the
levels. It is recognised that students' ability and capacity to
define fractions and explain equivalence are conceptually different
depending on where they are along the continuum of learning fractions
practised in classrooms and promoted by the syllabus.
Results revealed there is a tendency by some students to perceive
the double count of shaded parts and total parts as two unrelated
quantities. Attention needs to be paid to developing and consolidating
the notion of a fraction as representing a relationship between the two
Results also revealed students limited the idea of fractions to the
"part-whole" model. They linked fractions to pictures of
shaded parts of a model such as circles or rectangles and less
frequently to part of a group. To challenge and extend student
understanding, multiple contexts and representations should be used to
develop flexible interpretations and consolidate understanding of
The results highlighted the pedagogical importance of developing
students' conceptual understanding of the basic ideas of fractions
and equivalent fractions, namely, the part-whole relationship, number of
parts (partitions), and size of parts (units) using multiple
representations. Having students talk and write about how they create or
recognise equivalent fractions can strengthen their understanding and
provide valuable information to teachers (Niemi, 1996).
Kamii, C. (1994). Equivalent Fractions: An Explanation of their
Difficulty and Educational Implications. Accessed 7 September 2007 at
Cramer, K. & Henry, A. (2002). The development of
students' knowledge of fractions and ratios. In B. Litwiller &
G. Bright (Eds), Making Sense of Fractions, Ratios, and Proportions (pp.
3-17). Reston, VA: National Council of Teachers of Mathematics.
Kerslake, D. (1986). Fractions: Children's Strategies and
Errors--A Report of the Strategies and Errors in Secondary Mathematics
Project. Windsor: England: NFER-Nelson Publishing Company.
Larson, C. N. (1980). Locating proper fractions on number lines:
Effects of length and equivalence. School Science and Mathematics, 53,
New South Wales Board of Studies (NSWBOS, 2002). K-10 Mathematics
Niemi, D. (1996). Instructional Influences on Content Area
Explanations and Representational Knowledge: Evidence for the Construct
Validity of Measures of Principled Understanding. CSE Technical Report
402. Retrieved on 7 September 2007 from
Samtse College of Education
Royal University of Bhutan
University of New England
Table 1. Question 1: student responses.
Items Year 8(n=22) Year
Chosen Not Chosen Chosen
Two fifths 100% 0 100%
Two over five 63.6% 36% 66.7%
Two by five 0 100% 16.7%
Two upon five 4.5% 95.5% 25%
None of the above 0 100% 0
Items 6(n=12) Year 4(n=21)
Not Chosen Chosen Not Chosen
Two fifths 0 85.7% 14.3%
Two over five 33.3% 14.3% 85.7%
Two by five 83.3% 0 100%
Two upon five 75% 0 100%
None of the above 100% 0 100%
Table 2. Question 2: Student Responses.
Items Year 8(n=22) Year 6(n=12)
Chosen Not Chosen Chosen Not Chosen
Model A 90.9% 9.1% 83.3% 16.7%
Model B 54.5% 45.5% 66.7% 33.3%
Model C 72.7% 27.3% 83.3% 16.7%
Model D 0 100% 0 100%
Chosen Not Chosen
Model A 90.5% 9.5%
Model B 47.6% 52.4%
Model C 57.1% 42.9%
Model D 14.3% 85.7%
Table 3. Question 5: Student Responses.
Criteria Year 8 Year 6 Year 4
Represent and correctly explain 95.5% 83.3% 28.6%
Represent and incorrectly explain 4.5% 16.7% 71.4%
Table 4. Question 5: Correct and Incorrect Explanations.
Students Correct explanations Incorrect explanations
Year 8 "They are equal. This is "No they are not equal they
because the amount shaded are added (doubled) on."
in the same, it's just
broken up into different
fractions of each circle.
Also, 4/6, when simplified,
is 2/3, the same as the
"They are equal because the
area shaded is the same."
Year 6 "They occupy the same area. "No one of them has 2
Also you can see two sixths shaded and the other has
equal one third." "Yes, four shaded."
because 4/6 can be 2/3 by
going 4 / 2 = 2 and 6 /
2 = 3."
Year 4 "It is equal because it has "They aren't equal because
same amount but it has A has 2/3 shaded and B
been cut it into different has 4/6 shaded and they're
pieces. And another way of different fractions."
knowing to compare these
circles is that 1 piece of "No, because they have
the 2/3 is 2 pieces of the different numbers of pieces
4/6." shaded and they have
different pieces in total."
"In each circle the shaded "They are not equal because
part is the same area."
A has two out of three pieces
"The shaded parts are equal shaded and B has four out of
but one has smaller parts." six pieces shaded."
Table 5. Interview Question 1 Responses.
Description Year 8 Year 6 Year 4
Part of a whole J A D, E & F
Part of a number H - -
One number over another J - -
Does not know or could not say G - -
Figure 1. Question 1: Student justifications.
Year 8 "I chose A, B and C because I believe they represent 2/5.
A and C are the ones I have been shown before but B also
"A and C, because they are more clear, that two are only
coloured out of five."
Year 6 "The three I ticked showed five equal portions, of which
two portions are highlighted in someway."
Year 4 "There is five in each picture that I have chosen (A, B
and C) and two of them are shaded so it is two out of
"I chose this answer (A) because it is like a cake that
has divided into 5 and someone ate two and it is also easy
to know that it is 25 to me.'"