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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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