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Introduction

Students deepen their relational understanding

of the equals sign through exploring inequalities

in this competitive dice game, built around the

familiar fairy-tale e ree Little Pigs and e Big

Bad Wolf. e activity can be adapted to dierent

abilities by choosing more or less challenging dice

combinations. e two follow-up investigations,

based on the story Who Sank the Boat?, are intend-

ed to consolidate (Investigation 1), and further

extend (Investigation 2), student understanding

of the equivalence concept.

Context

Developing a relational understanding of the equals

sign involves students interpreting this symbol

as meaning ‘the same as’, rather than simply ‘the

answer’. It is a critical aspect of students’ develop-

ment in thinking mathematically that should be

promoted as soon as students begin encountering

number sentences (Karp, Bush & Dougherty,

2014). Such a relational understanding lays the

foundation for algebraic thinking and promotes

exible representations of numbers (Molina &

Ambrose, 2006).

For example, a relational understanding of

the equals sign supports ‘part-whole thinking’,

an important milestone in a young student’s math-

ematical development which involves the student

transitioning away from relying on counting-based

strategies to using partitioning and compensation

(Young-Loveridge, 2002).

is point is appropriately captured by Willis

(2000), in her description of two Grade 1 students

grappling with the number sentences 4 + 2 and 3

+ 3. Whilst Sam understands that he can use his

ngers to compute 4 + 2 = 6 and 3 + 3 = 6, for him

these facts remain unconnected bits of knowledge.

By contrast, Annie appears to grasp the connection

between them, which suggests the foundations for

an understanding of equivalence; in this instance

that 4 + 2 = 3 + 3 = 6. In her own words:

ey both equal 6 because if you take one

o the four and give it to the two, to make it

three, then it is 3 add 3 or you could take one

o the three and give it to the other three and

make 4 + 2. at’s why both have to be the

same. (Willis, p. 32–33)

Many mathematics educators view the frequent-

ly narrow conception of the role of the equals sign

in primary school classrooms as problematic. For

example, Perso (2005) argues that students are

conditioned to “do something now” or “nd an

answer now” whenever they encounter an equals

sign (p. 214). She contends that this action-

oriented, operational understanding of the symbol

prevents students considering its relational aspect,

which in turn impedes the development of

algebraic thinking. She suggests a range of peda-

gogical approaches for attempting to address this

misconception, including: using balance beams

to visually play with concepts of equivalence,

being exposed to practical worded problems which

encourage the use of compensation strategies,

Using picture story books

to discover and explore the concept of

James Russo

Belgrave South Primary School, Vic.

<mr.james.russo@gmail.com>

The notion of equivalence is a very important concept for students and should be developed from a

young age. This article demonstrates how students can deepen their relational understanding of the

equals sign by exploring inequalities within a dice game based on familiar children’s literature.

equivalence

26 APMC 21 (2) 2016

Using picture story books to discover and explore the concept of equivalence

and using partitioning to encourage students to

explore numerical equivalence in its symbolic form.

Despite its importance, developing this

relational understanding of the equals sign can

be extremely challenging, even when a teacher

spends considerable time exploring the concept

in the classroom (Seo & Ginsburg, 2003). One

possible means of laying the foundation for a deep-

er understanding of equivalence may be to provide

students with opportunities to discover this rela-

tional meaning of the equals sign. is discovery

can be promoted through juxtaposing the concept

of equivalence with the concept of inequality (and

the corresponding inequality signs) early in a stu-

dent’s mathematical development (Russo, 2015).

is article will introduce a competitive dice

game, built around the familiar fairytale, e ree

Little Pigs and the Big Bad Wolf, designed to foster

this discovery process. e article then outlines

two follow-up investigations based around the text

Who Sank e Boat? e rst investigation provides

students with a further opportunity to explore and

consolidate the concept of equivalence using a

dierent representation, specically the balance-

beam image suggested by Perso (2005). e

second investigation further extends the concept

of equivalence into a problem context involving

proportional reasoning.

The game:

Three Little Pigs

versus

The Big Bad Wolf

Teachers may wish to read a version of the fairytale

prior to the activity in order to engage students

before introducing students to the game.

Table1. Suggested dice.

Although the game is best suited to students aged

from six to nine, older children could still benet

from the activity.

Setup

Students should play the game in pairs. e only

equipment they need are various dice and some

paper and pencil (or a whiteboard and whiteboard

marker). e dice they should select depends on

the age group and current ability level of students.

e rules of the game are set out below:

• Rule1: In pairs, one student plays the pigs

and the other student the wolf.

• Rule2: Dice are rolled, and students calculate

their score for that role. For example, using

the Years 3–4 dice, the player representing

the pigs would sum the three 20-sided dice

together, while the wolf would halve whatev-

er number they rolled on their 10-sided 10s

dice. e player with the higher score records

the number sentence (using the greater-than

or less-than sign), and earns a ‘house’.

• Rule3: First to ve houses wins.

• Rule4: If both players obtain the same score,

they both record the number sentence (using

the equals sign), and both earn a ‘house’.

Teaching tips

It is recommended that rules 1, 2 and 3 be shared

with students prior to them playing game. ese

three rules should be presented unnumbered on

strips of card, and displayed prominently in the

classroom (see Figure 1).

If students have diculty during the game,

the teacher should refer the students to the three

game rules. However, rule 4 is best shared with

Suggested dice (with suggested operations in parentheses)

ree Little Pigs e Big Bad Wolf

Years 1–2 ree 6-sided dice

(add)

20-sided dice

(total)

Years 3–4

ree 20-sided dice

(add)

10-sided 10’s dice

(half)

27APMC 21 (2) 2016

Russo

Figure 1. Introducing the first three rules.

the students only once the game has commenced

or only after students raise the problem of players

obtaining the same score.

More specically, teachers should instruct

students in using the greater-than or less-than

sign appropriately in the pre-game introduction

when the activity is launched. For younger stu-

dents, consider introducing the application of

the greater-than or less-than sign as the “crocodile

always eating the larger amount”. It is recommend-

ed, however, that teachers do not provide students

with explicit instructions on what to do when the

scores are the same. Ideally, the teacher should

let the need to use the equals sign emerge from

students’ own reasoning, and explore this in

more depth during the post-game discussion

(see Figure 2).

If students ask about what to do in the case of

a tie during the pre-game discussion, the teacher

can respond something like “Hmmm I wonder if

that will happen? If it does, let me know and we

will decide what to do”. Obviously, if, during the

launch of the activity with the whole class, both

players obtain the same score, the teacher may

need to bring the discussion of rule 4 forward. e

teacher will need three rounds or so to demonstrate

the game to the students. It is worth noting that a

tie is relatively unlikely to occur. (Using the dice

recommended for older students, in a given round

the probability of a tie is less than 2%.)

Some questions for guiding the post-game

discussion appropriate for the rst (and, depending

on the age of the students, possibly second) time

students play the game include:

• What was the score in your game?

• Did anyone have both players roll the

same Wscore during a round?

• What did you decide to do?

Did the game rules help?

• What new rule do we need to include in

the game when both scores are the same?

Teachers working with older students (i.e., Years

3 and 4) can even get students to briey work on

this additional rule in pairs, record it and then share

it with the class. Rule 4 can then be introduced on

a strip of card, and displayed with the other rules in

the classroom.

Figure 2. Discovering the fourth rule.

Get students to play the game again in subse-

quent sessions using all four rules. e game, even

in this relatively simple format, can be revisited on

several occasions. If you feel that students require

further extension, the same basic game mechanism

can support the use of more sophisticated strategies

and concepts involving mental computation.

For example, try playing with ten little pigs (ten

6-sided dice) vs three big bad wolves (three 20-sided

dice); or, if exploring multiplication, three 6-sided

dice that need to be multiplied together (for the

pigs), vs a 10-sided 10s dice (for the wolf).

Example of a game

e game was played in a Year 3 and 4 composite

class using the appropriate dice as previously

described. Two Year 3 students Cada (pigs) and

Samantha (wolf) began a game together. On the

fth round, when the players already had two

houses each, Cada rolled a 20, 10 and 15 on her

20-sided dice, and Samantha rolled a 90 on her

28 APMC 21 (2) 2016

ten-sided dice. After Samantha halved the number

on her dice, the students realised that they had the

same score (see Figure 3).

As this was the rst time they had played the

game, a great deal of excitement followed, and Cada

yelled across the room “We got the same score, so

we don’t know which way the crocodile sign should

face. What should we do?! What should we do?!”

e teacher asked “What do you normally do when

two sides of a number sentence are the same? What

sign would you use?” Samantha replied elatedly

“e equals sign! ey are the same! We use the

equals sign!” e teacher replied that both students

could record the number sentence, and both earn

one house each. e need to use the equals sign only

arose in around one-third of the games, however

these instances provided a fascinating point of focus

for the post-lesson discussion (Note that playing

with the simpler dice, outlined for Grade 1 and 2

students, will result in the equals sign needing to

be used more frequently).

Figure 3. Example of a game: Cada (Pigs) vs Samantha (Wolf).

Consolidating and extending the

concept of equivalence:

Who Sank

The Boat?

Context

Read the classic children’s story Who Sank e Boat?

by Pamela Allen to the class, as a precursor to

launching the following investigations.

e rst investigation, “How can we balance

the boat?”, is designed to consolidate students’

understanding of equivalence. e investigation

explicitly incorporates Perso’s (2005) suggested

balance beam representation of equivalence and

allows students to tangibly and visually explore the

concept. e open-ended nature of the rst inves-

tigation, and its inclusion of an enabling prompt,

supports dierentiation and ensures it is a poten-

tially suitable activity for students in Years 1 to 4

(Sullivan, Mousley, & Zevenbergen, 2006).

e second investigation, “How heavy is the

mouse?”, is designed to build on the rst investiga-

tion (hence students should have already undertak-

en the rst investigation during a prior lesson).

It is considerably more challenging and is suitable

for older students (Years 3 to 5). It is designed

to extend student understanding of equivalence

through requiring students to apply the concept

to explore interrelationships between unknown

quantities. It involves proportional reasoning and

more closely resembles a formal algebraic problem.

Figure 4. Plasticine ‘models’ of the cow,

donkey, pig, sheep and mouse.

Using picture story books to discover and explore the concept of equivalence

29APMC 21 (2) 2016

Russo

Investigation 1: How can we balance

the boat?

Materials

• Paper and tape to create boats

• Playdough or plasticine to model the animals

• Pencils and paper to draw answers

Describing the problem

One of the reasons the boat stayed aoat so long

is because the animals worked out how to balance

their weights across the boat.

Can you nd a way to get all ve animals,

including the mouse, to distribute their weight

across the boat so that the boat is balanced and

stays aoat? Here is some important information

about the weight of the animals to help you with

the problem:

• e cow weighs the same as the donkey

(Cow = Donkey).

• e pig weighs the same as the sheep

(Pig = Sheep).

• e cow and the donkey are both heavier

than the pig and the sheep (Cow > Pig, Cow

> Sheep; Donkey > Pig, Donkey > Sheep).

• e pig and the sheep are both heavier than

the mouse (Pig > Mouse; Sheep > Mouse).

• See how many dierent ways you can solve

the problem.

What do the students need to do?

• Create a boat using paper and tape.

• Model the animals using plasticine

or playdough in accordance with the

above information (see Figure 4).

• Use their animal models and paper boat to

explore solutions to the problem (see Figure 5).

• Record their solutions by drawing them

on paper as they discover them.

Advice for teachers

• Encourage students to work in pairs or groups

of three to tackle the investigation.

Mathematical reasoning and critical thinking

can be supported by declaring that a solution

may only be recorded when all group members

agree that a particular conguration of animals

would balance the boat. If agreement cannot be

reached by the group on a particular congura-

tion, the teacher should consider photographing

it and exploring it further during the whole-

class discussion (it may provide an opportu-

nity to address a misconception or provide

an example where there is genuine ambiguity

about whether the boat would be balanced).

• ere are theoretically innite solutions to

this challenge, some of which are displayed

in Figure 5. However, the key insight into

the problem is realising that the mouse needs

to be exactly in the middle of the boat.

Enabling prompts for students

• If the mouse got onto the boat on his own,

where would he need to stand to balance

the boat?

• What if the mouse was in the middle of the

boat? Would this help you solve the problem?

Figure 5. Some possible solutions to the ‘How can we

balance the boat?’ investigation.

Investigation 2: How heavy is the mouse?

Materials

•Unixblocks

•Pencilsandpapertomodelanswers

30 APMC 21 (2) 2016

Describing the problem

Of course, the other reason the boat sank is because

the combined weight of the animals was too heavy

for the little row boat. You have been given some

extra information about the animals’ weights:

• e cow and the donkey are both twice as heavy

as the pig and the sheep (Cow = 2 Pigs, Cow =

2 Sheep; Donkey = 2 Pigs, Donkey = 2 Sheep)

• e pig and the sheep are both ve times

heavier than the mouse (Pig = 5 Mouse;

Sheep = 5 Mouse)

You have been told that the rowboat you have

made can hold up to 100 unix blocks before it

sinks, so the combined weight needs to be less

than this.

• Cow + Donkey + Pig + Sheep + Mouse

< 100 unix blocks

What is the maximum weight the mouse can be

(in unix blocks) to keep the boat aoat?

What do the students need to do?

Students can solve the problem however they like,

however the teacher may wish to encourage

students to physically model the problem with

unix blocks.

Advice for teachers

• e solution to the challenge is that the mouse

can weigh 3 unix blocks (i.e., 3 + 15 + 15

+ 30 + 30 equals 93, which is less than 100).

Although the challenge has only one solution,

the extending prompt is designed to get stu-

dents to generalise the relationships amongst

the variables (i.e., the animal weights), and

apply proportional reasoning. is process

can be viewed as constituting an elementary

form of algebraic reasoning (Perso, 2005).

• Although some concept of proportional rea-

soning is required to engage with the challenge,

students should be encouraged to pursue the

problem through trial and error. Combined with

the enabling prompt, this should provide many

students with a pathway into the problem.

Enabling Prompts

What if the mouse weighed one unix block?

How much would the pig and sheep weigh?

What about the cow and donkey? How much

weight would there be in the boat altogether?

Extending Prompts

Work out the maximum weight the mouse can

be if the boat can hold up to:

• 200 unix blocks

• 300 unix blocks

• 500 unix blocks

• 1000 unix blocks

• 10000 unix blocks

Conclusion

Building a deeper understanding of equivalence,

and, in particular, grasping its relational aspect is

both critical to developing number sense (Karp et

al., 2014) and potentially very challenging (Seo &

Ginsburg, 2003). It is suggested that playing the

ree Little Pigs dice game, and undertaking the

follow-up investigations using the text Who Sank

e Boat? can help students to engage authentically

with this critical concept.

References

Allen, P. (1982). Who sank the boat? Australia: omas

Nelson.

Karp, K. S., Bush, S. B., & Dougherty, B. J. (2014).

13 rules that expire. Teaching Children Mathematics,

21(1), 18–25.

Molina, M., & Ambrose, R. (2006). Fostering relational

thinking while negotiating the meaning of the equal sign.

Teaching Children Mathematics, 13(2), 111–117.

Perso, T. (2005)in M. Coupland, J. Anderson, & T. Spencer

(Eds.) Making mathematics vital: Proceedings of the

twentieth biennial conference of the Australian Association

of Mathematics Teachers (pp. 209-216). Sydney, Australia:

AAMT.

Russo, J. (2015). Surf’s up: An outline of an innovative

framework for teaching mental computation to stu-

dents in the early years of schooling. Australian Primary

Mathematics Classroom, 20(2), 34–40.

Seo, K. H., & Ginsburg, H. P. (2003). “You’ve got to

carefully read the math sentence...”: Classroom context

and children’s interpretations of the equals sign. In A. J.

Baroody & A. Dowker (Eds.), e development of arith-

metic concepts and skills: Constructing adaptive expertise

(pp. 161–186). Mahway, New Jersey: Lawrence Erblaum

Associates, Publishers.

Sullivan, P., Mousley, J., & Zevenbergen, R. (2006). Teacher

actions to maximize mathematics learning opportunities

in heterogeneous classrooms. International Journal of

Science and Mathematics Education, 4(1), 117–143.

Willis, S. (2000). Strengthening numeracy: Reducing risk.

Paper presented at the Australian Council for Educational

Research (ACER), Improving numeracy learning: Research

conference 2000: Proceedings. 31–33.

Young-Loveridge, J. (2002). Early childhood numeracy:

Building an understanding of part-whole relationships.

Australian Journal of Early Childhood, 27(4), 36.

Using picture story books to discover and explore the concept of equivalence

31APMC 21 (2) 2016