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The power of a single game to address a range
of important ideas in
fraction learning
DOUG CLARKE & ANNE ROCHE describe a fraction game used to develop key concepts including equivalence and addition of fractions.
As part of the Contemporary Teaching and Learning of Mathematics Project (CTLM1), the mathematics education team at Australian Catholic University has the privilege of working with principals, teachers, students, and parents in schools in the Melbourne Archdiocese.
A particular highlight is the opportunity to work alongside project teachers and their students in developing worthwhile and relevant classroom activities. In the last 3 years, we have had the chance to try out particular fraction and decimal lessons, making subtle changes each time, as we and our teacher colleagues learn from each classroom experience.
18 APMC 15 (3) 2010
1 CTLM is funded by the Catholic Education Office (Melbourne). We acknowledge gratefully the support of the CEOM for our work, and the collaboration of our teacher colleagues in the project schools.
The Power of a Single Game to Address a Range of Important Ideas in Fraction Learning
In this article, we will describe one such activity, Colour in Fractions, including its evolution, the mathematics which it has the power to address, and some hints for teachers using it for the first time. We mentioned this activity briefly in Clarke, Roche and Mitchell (2007), but have gradually refined it, and in this article, we provide a detailed outline of the purpose and benefits of its use.
Fractions: Difficult to teach and to learn
Fractions are widely recognised as a very important topic in the middle years, but one that is difficult to teach (Ma, 1999) and to learn (Behr, Lesh, Post & Silver, 1983). Among the factors which make rational numbers in general and fractions in particular difficult to understand are their many representations and interpretations (Kilpatrick, Swafford & Findell, 2001).
Many teachers complain, with justification, that we have a crowded mathematics curriculum in the middle years, leading to a strong tendency to treat topics at a "surface level", given the pressure to "get through it all." Our experience is that state/territory and school curricula and consequently teachers give inappropriate and premature attention to the four operations with fractions and decimals, while failing to give appropriate emphasis to more important foundational notions, such as, for example, fraction as division, fraction as operator, and fraction as measure (Clarke, 2006; Clarke et al., 2007).
Given the pressure of the crowded curriculum, it is exciting, therefore, to find games/lessons/activities which address a number of important ideas in a challenging but accessible and enjoyable way.
The data from 323 interviews with students at the end of Year 6 (Clarke, Roche, Mitchell & Sukenik, 2006) indicated that students need classroom experiences which assist them to understand more clearly the roles of the numerator and denominator in a fraction, the meaning of improper fractions,
and the relative sizes of fractions. The game discussed in this article has been shown to provide assistance in all of these areas and others.
Teachers have used fraction walls of various kinds for many years in teaching fractions, and in many cases, played games of a similar kind to the one below. However, it is some of the refinements to the game which in our opinion have made the impact more substantial.
Colour in Fractions: The rules
Students have dice that create fractions up
to twelfths, and a fraction wall. They colour
in sections of the wall that correspond to the
fractions that they roll with the dice. They
have:
? one die labelled 1, 2, 2, 3, 3, 4 in one
colour
? another die labelled * , * , * , * , * , * in
another colour
2 3 4 6 8 12
? a fraction wall as shown in Figure 1.
Figure 1. Fraction wall
Each horizontal strip is one whole. So,
the first strip is made up of two halves, the
next one three thirds, and so on. Players in
turn throw both dice. They make a fraction,
the first die being the numerator. They then
colour the equivalent of the fraction shown.
For example, if they throw 2 and * , then they
can
colour
in
2
of
one
line,
or
4
4
of
one
line,
or
1
4
of one line and
2
8
of another, or any
4
8
other combination that is the same as
2.
4
If a player is unable to use their turn,
they "pass." The first player who colours in
their whole wall is the winner, but the other
player is encouraged to keep going (with
the support of the first player) to fill their
fraction wall, if time permits.
APMC 15 (3) 2010 19
Clarke & Roche
How we introduce the game to students
We believe that the game as described here is
appropriate for an introduction to fractions
from Grades 4 to 8, as different students will
gain more from the game each time they
play. In fact, we believe strongly that teachers
should encourage students to play the game
in full at least three times, in order for its
potential to be realised.
In introducing the game, we make use of
an A3 version of the fraction wall, stuck to
the board, and gather the class around to
demonstrate. We invite one student to roll
the two dice and generate a fraction. Because
we need a fraction that has a number of
straightforward equivalences, we prefer
1
4
something
rolls
1 12
or
l4ik,efo4r
3
or 8 , and if the student example, we ask them to
roll again. We choose not to demonstrate
the consequences of rolling an improper
fraction, as we prefer this to arise naturally
during the game.
Assuming
that
1 4
has
been
created
(say),
we ask the student what they choose to
shade in. The student usually shades in
one of the quarters, but we ask the class to
consider
other
possibilities,
and
usually
2 8
is
suggested. We point out that they can shade
in any fraction equivalent to one strip and part of another
14(,e.egv.e, n*614
part of
+
1 12
).
Of course, once they decide, they must only
shade in one equivalence. We also mention
that they do not have to start from the left of
the fraction wall each time. Please note that
the second player does not do anything with
her/his board as yet. The second player's
turn will be next.
We then show them the fraction wall and
suggest the following:
Each roll, the student should use a different
colour pencil or texta. This means that it is
easy for them and for the teacher moving
around to follow clearly the decisions made
at each stage of the game. The latest version
of the game sheet has "what I rolled" and
"what I shaded" columns, so that these can be
distinguished and considered at a later stage.
For example, ", but shaded
a student
"
1 4
+
1 8
".
A
might
have
rolled
"
3 8
sample game sheet is
shown in Figure 2.
Figure 2. A sample game sheet.
The game then proceeds as follows:
1. They should take it in turns to roll and
shade, and if the fraction rolled or its
equivalence cannot be shaded, they miss
a turn. This becomes more frequent later
in the game.
2. They are not allowed to break up a "brick."
3. In finishing off the game, the student
must roll exactly what they need. A larger
fraction is not acceptable to finish. So, for
eroxlalm2p(lec,leiaf rtlhyemy onreeedth18antoisfinneisehdeadn)d,
they they
4
must miss a turn.
Naming and representing improper fractions: A powerful opportunity provided by this game
As will be evident to the reader, no mention has been made to date of improper fractions, which are bound to arise at some stage in the game, because the dice are designed to make this happen. Although some teachers assume that a lesson on improper fractions should precede the game, our experience is that it is better to just let improper fractions arise naturally, and then look at how students choose to deal with them.
20 APMC 15 (3) 2010
The Power of a Single Game to Address a Range of Important Ideas in Fraction Learning
We have noticed that some students in
the
middle
years
who
roll
4 3
call
it
"four
threes." This use of whole number rather than
fractional language appears to be an indicator
that the students do not yet understand which
digit refers to the number of parts or the size
of the parts. This provides the chance with
the individual or the whole class to draw their
attention to the meaning of the numerator
and the denominator in a fraction. We see
this as a key teaching point within the lesson.
When students are first trying to make
sense of common fractions, teachers have
typically defined them as follows:
"The denominator tells you how many
parts the whole has been broken up into, and
the numerator tells you how many of these
parts to take, count or shade in."
Now, this works reasonably well for
fractions between 0 and 1, but not well
for improper fractions. We prefer this
explanation for students:
"In the fraction
a b
,
b is
the name or
size
of
the part (e.g., fifths have this name because
5 equal parts can fill a whole) and a is the
number of parts of that name or size."
So
if
we
have
4 3
,
the
three
tells
the
name
or size of the parts (thirds), and the 4 tells us
that we have 4 of those thirds (or 1 13).
With this explanation, students quickly see
that rolling an improper fraction, early in the
game, can work to their advantage
playing the game in mixed ability pairs is that
students will learn from each other as they
play, gradually increasing their understanding
of possibilities.
At the conclusion of the game, we
often ask students to share with the class
their most interesting turn and describe
what they rolled and what they shaded.
Reading straight from their game sheet they
can
recall
a
roll
(say,
3 4
)
and
then
describe
the bricks that they chose to shade for that
roll (e.g., 2 + 1 + 3 ). This sharing of these
choices
8
will
als4o
12
contribute
to
the
learning
that occurs with peers.
Rather than waiting for one player to
completely shade the fraction wall, teachers
sometimes find that the game may
be better concluded at a specified
time, and the players can then consider
who will be the winner--the player
whose fraction wall has the least total yet
to be shaded. Determining the winner in
this case is an interesting mathematical task
in itself.
At some stage, the teacher may wish
to point out that if the fraction wall is
completely shaded, then all of the shaded
parts must add up to 6 (six wholes), or
alternatively draw this from the students.
Students can be invited to check this for
their game.
A discussion of strategies Accessibility of the game to all students
In pulling the lesson together, we often ask
One of the features of the game is its students to consider one of two questions:
suitability for students with a range of 1. If you played the game tomorrow, what
levels of understanding of fractions. Some students will roll 3 and shade in three of the
4
quarters if the quarters are as yet unshaded,
would you do differently? 2. If you were giving some hints to a younger
brother or sister who was about to play the
without considering any other options. If they
have already "used" the quarters and they roll 3, they will elect to miss a turn. However,
4
students who have a good understanding
game, what would these hints be? It is interesting that students' hints often relate to either shading the little bits first (e.g., the twelfths or the eighths) so that they
of equivalence will of course look for a are not "stuck" with them at the end, or to
range of equivalences of 3 and choose one shading the large parts first, leaving them
4
which suits the situation. The advantage of with more possibilities at the end.
APMC 15 (3) 2010 21
Clarke & Roche
Where's the mathematics? A quick summary of the mathematical potential of the game
Given the crowded curriculum, it is important
to realise the many aspects of fractions
addressed during the game. Teachers identify
the following, with the first four being the
major points:
? Equivalent fractions--the physical
representation of the fraction wall
enables students to "line up" particularly
difficult fractions to generate equivalent
4
combinations (e.g., lining up
same
as
1 6
and
2 12
combined).
12
as
the
? This game encourages the use of
"fractional language" (e.g., three-quarters
instead of three out of four) which is
helpful for understanding fractional parts
within the part-whole construct.
? Understanding improper fractions--the
game provides an excellent introduction
to improper fractions, in a context where
understanding is motivated by the need
to use them to advantage in the game
context.
? Addition of fractions--the game provides
a very appropriate introduction to this
notion, as students record their various
sums in the "what I shaded" column, and
notice some of the patterns involved.
? Problem solving--as there are many
options possible at most stages of the
game, the students have to weigh all of
these up, before committing to a particular
shading.
? Visualisation--sometimes the students
are challenged to see how what they
have rolled might be represented by a
combination of some of the remaining
parts--this can be quite a visual challenge
on occasions.
? Probability--what is the chance that I will
get what I need? As the teacher moves
around the room, they can ask individuals
or pairs, "What do you still need and
how might you get this?" For example, a student might still need 2 and could get
12
this as either
1 12
to
finish,
it
1
6
is
or
2 12
.
If
they
only
need
a good chance to discuss
the fact that they would only expect on
average to form this fraction one in every
36 rolls (one chance in 6 on one die--the
"1"; and one chance in 6 on the other die--the " * ).
12
Follow up investigation--each group gets a fraction
After a couple of lessons playing the game,
each group could be given a particular
fraction (e.g.,
3 4
,
4 6
,
4 3
)
and
invited
to find
as many equivalences as they can, which are
possible with the fraction wall in the Colour in Fractions game. A sample small group
response is shown in Figure 2.
Figure 3. Expressions equivalent to 4 . 3
Another challenge is to present the
students with a hypothetical game in progress,
and invite them to suggest the way in which a
particular roll might work out for the player.
For example, children could be shown the
game in progress depicted in Figure 4 , where
13 ,
21 162
,
2,
8
and
1 12
remain
unshaded.
They are
then told, "Imagine someone has just rolled
the
fraction
2 3
(or
3 8
or
4 12
or
1
2,
respectively).
What would you recommend they shade in,
or do they have to miss a turn? In particular, is
it possible to complete the fraction wall with
one more roll of the dice?" (As the reader will
determine, the answer is "Yes.")
22 APMC 15 (3) 2010
The Power of a Single Game to Address a Range of Important Ideas in Fraction Learning
Figure 4. Example of a game in progress.
An approach for students who find the creation of equivalences a challenge
One teacher told us about a way she makes
the game accessible to students who find
the equivalences a great challenge. She gives
the students a clean fraction wall and asks
them to label all the bricks (each one of
the halves
1
as
1 2
,
each
one of the
thirds
as
3, etc.). They then cut out all the pieces and
when they are playing the game and they roll
3 4
for
example,
they
some of the bricks to
physically manipulate
carefully
match the
3 4
,
and, having found one that works, they then
complete their turn on the game sheet.
In summary: A game with great potential
References
Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In M. Landau (Ed.), Acquisition of mathematics concepts and processes (pp. 91?126). Hillside, NJ: Lawrence Erlbaum Associates.
Clarke, D. (2006). Fractions as division: The forgotten notion. Australian Primary Mathematics Classroom, 11(3), 4?10.
Clarke, D. M., Roche, A., & Mitchell, A. (2007). Year six fraction understanding: A part of the whole story. In J. Watson & K. Beswick (Eds), Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 207?216). Hobart: MERGA.
Clarke, D. M., Roche, A., Mitchell, A., & Sukenik, M. (2006). Assessing student understanding of fractions using task-based interviews. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds), Proceedings of the 30th Conference of the International Group of Psychology of Mathematics Education (Vol. 2, pp. 337?344). Prague: PME.
Kilpatrick, J., Swafford, J., & Findell, B. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' knowledge of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.
Roche, A. (2010). Decimats: Helping students to make sense of decimal place value. Australian Primary Mathematics Classroom, 15(2), 4?12.
Doug Clarke & Anne Roche Australian Catholic University (Melbourne)
APMC
Hopefully, we have outlined the potential of a game in assisting students to develop key concepts of equivalence and addition of fractions. We invite the reader to try out the game with its various enhancements with a group of students at least three times, and let us know how it goes. The reader may also be interested in the Decimat game (Roche, 2010), which uses a similar model and game to assist students' developing understanding of decimal fractions.
APMC 15 (3) 2010 23
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