RIC-6001 Developing mathematics with pattern blocks

CONTENTS 

Contents ....................................................................................... i 

Foreword ...................................................................................... ii 

Pattern blocks – An aesthetic mathematical experience ............ iii 

A guide to using pattern blocks in the classroom ....................... iv 

Challenging the step-by-step approach ..................................... v 

Introducing Learning Centres ...................................................... vi 

Towards mathematical abstraction ............................................ vii 

Becoming familiar with pattern blocks ........................................ 2 

Know the blocks .......................................................................... 3 

Pattern Block Man ................................................................... 4–5 

Pattern blocks games ............................................................. 6–7 

How old is Pattern Block Man? ............................................... 8–9 

Fences and floors ...................................................................... 10 

Expanding the shape experience .............................................. 11 

Picture puzzles .......................................................................... 12 

Fill the shapes ..................................................................... 13–15 

Fractions .............................................................................. 16–33 

Picture values ...................................................................... 18–19 

Cover-ups ............................................................................ 20–21 

Fractions – 1 ........................................................................ 22–23 

Fractions – 2 ........................................................................ 24–27 

Giving the shapes value ...................................................... 28–31 

Pattern block fractions ........................................................ 32–33 

Space, shape and spatial relationships ............................... 34–65 

Enlarging shapes ....................................................................... 36 

Comparing shapes .................................................................... 37 

Mirror, mirror ........................................................................ 38–39 

More mirrors ........................................................................ 40–41 

Exploring angles .................................................................. 42–44 

Seen from all directions ............................................................. 45 

No overlaps ......................................................................... 46–47 

Symmetry ............................................................................ 48–49 

Reflection pictures ............................................................... 50–51 

Mirror symmetry .................................................................. 52–53 

Round and round ................................................................ 54–55 

New shapes from old ................................................................ 56 

Hexiamonds .............................................................................. 57 

Making triangles .................................................................. 58–59 

Exploring hexagons ............................................................. 60–61 

Perimeter ............................................................................. 62–63 

Perimeter and area .................................................................... 64 

Perimeter patterns ..................................................................... 65 

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Trading games ..................................................................... 66–73 

All yellow ............................................................................. 68–69 

Trade with pattern blocks .................................................... 70–71 

Shopping with pattern blocks ............................................. 72–73 

Shared ideas ............................................................................. 74 

Photocopiable resources .................................................... 75–78 

All-block grid ............................................................................. 76 

Triangle grid ............................................................................... 77 

Square grid ................................................................................ 78 

Hexagon grid ............................................................................. 79 

Answers ..................................................................................... 80 

Glossary .................................................................................... 81 

i

ii 

ii 

Foreword 

For many decades, the use of manipulative materials 

to assist young children in their learning of mathematics 

has been recommended. The advocacy of great 

educators such as Maria Montessori, Zoltan Dienes and 

Catherine Stern encourages a wide acceptance of the 

use of manipulative materials, especially in primary school 

classrooms. Once, it was felt that simply giving students 

manipulatives to use in mathematics lessons would be 

enough to develop an understanding of mathematical 

concepts. This is not true. Manipulatives in and of themselves 

do not teach—skilled teachers do. 

This series—Hands-on mathematics—is designed to help 

teachers who are trying to make the most of students’ 

experiences with manipulatives. We believe it is better to use 

a few well-chosen products rather than an array of ‘bits and 

pieces’. We recommend ‘a lot of a little’ rather than ‘a little of 

a lot’ when it comes to working with manipulatives. It is better to 

focus on a few well-chosen manipulative materials so that students 

will have an adequate supply of pieces. Nothing is more frustrating 

than not having enough to finish ‘creating a design’ or ‘building that 

masterpiece’. As well, it is important that sufficient materials are 

available to allow models to be left on display in the classroom. 

Frequently, when we work with students and teachers in classrooms 

and workshops, a range of common concerns is raised. Let us share 

a few with you. 

Why use manipulatives? 

When used as part of a well thought-out lesson, manipulatives can help 

students ‘come to grips’ with difficult concepts. The key to good use of 

manipulatives is for teachers to have a clear goal in mind when using them. 

This will help maintain the intention of the lesson and focus responses to 

any questions asked during the lesson. Teachers will have a clear idea of 

what to look for when observing students using manipulatives. 

As Richard Skemp, the famous educational psychologist said, ‘It is as though 

their thinking was out there on the table’. 

We have observed how students experiment with ideas willingly. If, at first, 

satisfaction with an idea is not achieved, students will seek another solution. We 

do not see this happening as frequently when students are expected to work with 

abstract statements such as equations and written problems. 

The skilled use of manipulatives—note, we said the skilled use of manipulatives— 

will enhance mathematics outcomes. Poor use may be detrimental to student 

attainment. This series of books is designed to ensure skilled use of manipulatives 

in the classroom. 

Is there a difference between a mathematics manipulative and a 

mathematics teaching aid? 

We believe there is a big difference between the two types of materials. 

In fact, actively engage with the students as their thought 

processes emerge. Simply using manipulatives is not enough. 

Students need to be given time to reflect on their activity and 

share their thoughts with a group or the whole class. The 

teacher plays a vital role in helping students connect new 

knowledge with old. Language plays a key role throughout 

this learning process. 

What evidence can I show that students are learning 

or have learnt…? 

Some teachers are concerned about the lack of written 

evidence to substantiate learning when manipulatives 

form a large part of the lesson. There are several ways a 

student might record his/her findings: 

• writing about the experience 

• sketching or drawing any models produced 

• photographing any models produced 

• presenting ‘learning tours’ to students in other 

classrooms 



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A child can interact, even take control of a good mathematical manipulative; whereas a 

teaching aid tends to control the learning experience. Too often, a teaching aid is used 

as a ‘telling’ support rather than a learning support and experience has taught us that 

‘telling’ is not a very successful method of teaching mathematical ideas. 

How will I know whether the students are learning anything? 

Observe the students as they work with the manipulatives. Don’t worry if they solve a 

problem in a way different from what you expected. Ask questions. Encourage students to 

explain their thoughts or write about their experience. 

• maintaining a learning journey logbook. 

Actually, when preparing this type of learning 

evidence, students have a wonderful opportunity to 

reinforce their own learning. 

How do I manage the use of manipulatives? 

Some teachers worry that students will only play 

with the manipulatives and not pay attention, 

or worse still begin to throw the material 

around. These are genuine fears which will 

decrease as experience, both by the students 

and teacher, increases. As with any ‘new toy’ 

there will be a ‘novelty effect’. The first time 

you introduce a manipulative, allow time for 

the students to explore. 

Set some simple rules and limits for the 

way the material is used and enforce 

these early on. Students will soon learn 

to respect the material. 

Throughout this book, management 

ideas are presented. We encourage 

you to adopt them as your own.

Pattern blocks 

An aesthetic mathematical experience 

A set of pattern blocks is made up of the following six pieces: 

The shapes are designed so that all the sides are the same length, except the base of the red trapezium 

which is twice the length of the other sides. The specific properties of each shape are outlined below. 

Yellow hexagon 

The hexagon is a regular hexagon 

which means all the sides are the 

same length, all the angles are the 

same size (congruent) and each 

angle is 120º. Opposite sides are 

parallel. 

Orange square 

A square is a regular quadrilateral. It is 

made up of four sides of equal length. 

Opposite sides are parallel. All four angles 

are right angles (90º). The diagonals are of 

equal length and bisect each other at right 

angles. 

Red trapezium 

A trapezium is a quadrilateral with one pair 

of parallel sides. The trapezium used in the 

pattern block set is a special trapezium; that 

is two sides are the same length. The base is 

twice the length of the standard side length. 

Blue rhombus/Tan rhombus 

A rhombus is a quadrilateral (a shape with four 

sides) that has four sides of equal length. Opposite 

sides of the rhombus are parallel. Opposite angles 

are the same size. The diagonals bisect each other 

at right angles. Note the sum of the angles in a 

quadrilateral is 360º—a quadrilateral may be divided 

into two triangles. 

60º 

120º 

120º 

120º 

120º 

120º 

120º 120º 

120º 

60º 60º 



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120º 

120º 

30º 

60º 

150º 

150º 

30º 

Green triangle 

The green triangle is an equilateral triangle, which means 

all the sides are the same length. All the angles are the 

same size, 60º. Note that the sum of the angles in a 

triangle is 180º. 

60º 60º 

60º 

Pattern blocks are available in a wide range of materials. Be careful as you select the 

materials you wish to use with your students. We believe it is important that the blocks 

are accurately cut and have ‘body’ or depth. As well, we recommend you acquire 

blocks that use the standard colours. 

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iv 

On their own, pattern blocks 

will not teach very much at all. 

In fact, they are no more than 

a catalyst—a supporter in the 

learning operation. It was Maria 

Montessori who said the hand is 

the chief teacher of the student. 

We will take that a little further 

to say that it is the senses which 

provide the data on which the 

student will learn. A teacher 

helps facilitate that learning. 

Likewise, pattern blocks will 

aid the learning process. 

Keep the containers and 

blocks clean 

Manipulative materials are as important 

as any other teaching materials. Take 

pride in the manipulative materials you 

have in your classroom. They provide 

important impetus to a student’s 

mathematical development, hence 

learning. 

A guide to using pattern 

blocks in the classroom 

Sufficient quantities 

We recommend that a classroom has access to at least eight containers of 

250 blocks (one per group of 4) at any one time. Students relax when they 

know there is sufficient material available. As well, that permits models to be left 

on display for extended periods. Some teachers place a collection of pattern 

blocks in drawstring bags or plastic containers to speed up distribution in the 

classroom. 

Limit the length of time the students use the pattern blocks. 

The idea of 2000 pattern blocks is not really extravagant because the blocks 

need to be in the classroom for about three weeks at a time. In a 10-week 

term, that means three classrooms can have exclusive use of the material for 

three straight weeks. After three weeks, another set of materials, for example, 

Unifix ® cubes, can be rotated into the classroom for a three-week period. This 

system works very well. Our motto: Three weeks on: six weeks off. 

During the six-week break, the student’s brain carries out a great deal of 

assimilation. When the pattern blocks return, the students are ‘raring-to-go’ 

once again. The same comment can be made about other materials. The 

break is most beneficial. 

Using materials 

efficiently in a school 

Some schools have three Year 2 

classes. By circulating materials 

on an organised basis, we 

need three realistic amounts of 

materials—say, Unifix ® cubes, 

pattern blocks and Base Ten 

blocks —enough in each collection 

for use in a whole Year, stored 

in appropriate containers and 

accompanied by satisfactory 

teacher resource ideas. At the end 

of each three-week session, the 

tubs are circulated to the other 

classrooms. Some schools carry 

out a two-week circulation routine, 

but experience has shown that 

three weeks is a better period: it 

allows more time to ‘wind-up’ the 

ideas presented. 



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Easy access at all times 

Store the pattern blocks in plastic 

tubs that can be carried easily. 

Keep them in a dedicated space 

in the classroom so students 

can have access to them at any 

time. Models of solutions and 

creations can be displayed in easily 

accessible positions. While a school 

is in operation, the blocks will be in 

classrooms, so no special storage 

facilities are needed. This means the 

manipulatives are out in classrooms, 

where they belong, rather than 

collecting dust in the storeroom. 

Take-home bags 

Some schools encourage 

students to take home a 

drawstring bag containing 

50 or so pattern blocks 

instead of a reading book. 

A small card may explain an 

activity which the student 

teaches the parents. 

So, rather than a parent 

demanding, ‘What did you 

do in mathematics today?’, 

a student can show the 

blocks and say, ‘Look, I can 

do fractions’. 

Do not fear you will lose the 

pattern blocks. The students 

will have developed a pride 

in them and will take very 

good care of the pattern 

blocks.

The mathematical world of a child 

does not develop in a straight line or 

a predictable sequential manner. In fact, 

it could be said the whole procedure 

is pretty messy. Ideas shoot into the 

brain from all directions in no real 

coherent order. It is the function of the 

brain to find an order which suits the 

child at that time. Add more ideas and 

that order will be re-shunted. It is an 

ongoing process. 

Here seems a paradox. As educators, 

we know that for most learning to take 

place certain stages of development 

need to have been achieved by a human 

being. It seems that development is 

independent of our learning ideas. Some 

would say there is no point in attempting a 

certain piece of learning unless the learner 

has reached a certain stage of development. 

Others would suggest that providing learning 

experiences before the child is ready could cause 

long-term damage to the child’s capability and 

willingness to learn. But so many of our text 

resources and curriculum documents find this hard. 

Challenging 

the step-by-step approach 

As a teacher there will be times when you will 

find the ideas you want to be learnt are only 

half understood and you become frustrated 

and perhaps even say things like, ‘That child is 

not learning’. Maybe that child is not ready to 

learn what you want him/her to learn! Suddenly, 

after sharing some other activities and, often, 

on some other day, that child will demonstrate 

the ‘aha!’ factor—’I understand’. Why should a 

young child’s experience be any different from 

yours? You get ‘ahas’ any time and in any place. 

When preparing a text such as this, ideas can 

be presented in a carefully laid out plan. But this 

does not mean that the student learns these 

things in that order and it definitely does not 

mean that you have to present the ideas in the 

order offered here. 

All these ideas have been well tried with 

children and their introduction to the young 

learners can be just another part of their reallife 

experiences. Give the students a chance to 

create a whole picture of mathematics, rather 

than force isolated sections on them. 



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v

Introducing 

Learning Centres 

We encourage the establishment of Learning Centres in a classroom 

to promote the development, hence strengthening, of ideas. In 

groups of three to six, class members may congregate around a 

challenge and spend their time on the idea: let’s not call it a task. 

Imagine! At any one time in your 

classroom several widely different 

activities will be involving the 

students. We have seen this type 

of classroom organisation used 

in many primary schools. An interesting benefit of this classroom 

management technique is that the students take ownership of the 

Learning Centres. And, from your point of view, you will be very 

satisfied with the amount of ‘good learning’ taking place. 

Students need time to 

experiment with ideas 

You can integrate activities from other 

curriculum areas. A colleague organised all 

his lessons using Learning Centres as the 

foundation of his classroom management. 

Each morning, he greeted the members 

of the class and frequently presented a 

lesson to the whole group. Often, this 

lesson served to initiate a new Learning 

Centre. At any time, there were about eight 

Learning Centres in the room, three or 

four of them being introduced during the 

week. Sometimes, Learning Centres were 

removed and then ‘re-installed’ weeks later 

to help reinforce learning experiences. 

Key elements in a Learning Centre 

• Students keep 

a record of their 

activity in a simple 

Learning Centre 

logbook. We do not 

insist on copious 

records: reminders 

are written to 

encourage 

conversation. The 

maintenance of the 

classroom Learning 

Centres is the 

responsibility of the 

class members. 

Some students 

may find it difficult 

to interpret the 

instruction card but 

will find classmates 

willing to help out. 



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• An activity space or a 

table or pair of desks 

may be dedicated to 

a Learning Centre. 

We often cover the 

activity space with a 

bright tablecloth. It 

looks good and the 

students like it. 

• Appropriate manipulative 

materials are kept at the 

Learning Centre. These 

will include the correct 

manipulative, recording 

materials (paper and 

pens) and (very) simple 

instructions. 

• Throughout this book, pages 

have been dedicated to 

Learning Centre ideas. Simply 

photocopy then laminate the 

page and display it on card 

holders. Students learn very 

quickly to ‘obey’ the ‘number’ 

rule, which indicates the 

maximum number of students to 

be at a Learning Centre at any 

one time. 

• The teacher’s role 

is transformed 

from one of upfront 

director to 

one of a sharing 

participant. 

vi

Towards 

Zoltan P Dienes, the famous 

mathematics educator/psychologist, 

saw that children acquired 

understanding gradually and only 

after sufficient directed play had been 

experienced. Many other mathematics 

educators, like Richard Skemp, have 

noted the need for the same experiential 

approach. Today, this approach may be 

known as constructivism, the basis of 

which assumes that a learner shapes 

his/her learning via interaction with 

the environment. The teacher has 

a vital role in shaping that learning 

environment. 

No matter what activity you present to 

the students, the stages of ‘Discover’ 

and ‘Talk’ will be experienced. Students 

need time to ‘find out what it is all 

about’. For some, it may be a new 

experience altogether. Consequently, 

more time will be spent at the Discover 

and Talk stages of development. In 

contrast, some students may be well on 

the way to formalising a concept; hence 

they will be engaged at the ‘Explain’ 

and ‘Symbol’ stages. 

We have observed students passing 

through stages of mathematical 

understanding and capability. We 

have adopted an initialism for this 

sequence—DTES—and have applied 

it throughout this collection of ideas 

on how to best use Pattern blocks. 

Where possible, for your guidance, 

we have indicated an approximate 

developmental stage for various 

activities. 

Using DTES as a guide 

On various activities we will 

use the DTES symbol to 

provide a rough guide of the 

developmental levels involved 

in the activity. 

mathematical abstraction 

Discover 

A child experiences his/her environment: the child 

sees, hears, feels, tastes, smells and handles. In 

fact, all learning begins at this point. The wider 

the experience, the richer will be the language 

development. 

A teacher does not need to direct this experiential 

stage—there is no harm in suggesting ideas and, 

if the students ‘run’ with them, good! On the other 

hand, this is a time when the teacher will be able 

to observe the students, note their developmental 

stages, and talk to them about their ideas. 

Do not rush this stage! 

Symbol 

Within the written symbol, there 

is a huge amount of knowledge 

(discover, talk and explanation). 

To make this symbol and to 

comprehend other symbols is a 

very sophisticated achievement 

and must never be forced or ‘fastforwarded’. 

D T E S 

Talk 

A child develops the spoken 

language to describe and 

communicate that experience. 

The interaction with peers and 

significant others, such as parents 

and teachers, strengthens that 

development. 

Explain 

There are many ways in which 

children will explain their ideas— 

perhaps in speech, pictures, 

writing or actions. But, whatever 

the type of representation, the 

child recognises the association 

with the original ideas. 



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D T E S 

Students have 

developed many 

conceptual ideas. 

Now, they are ready to 

present them formally. 

D T E S 

Students will ‘lead’ 

the activity as they 

discover various 

ideas and suggest 

conclusions. 

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D T E S 

Becoming familiar 

with pattern blocks 

Purpose 

To become familiar with the 

pattern block pieces. 

Background 

Pattern blocks come as a 

set. The set is made up of six 

shapes: 

• equilateral triangle...... green 

• square....................... orange 

• rhombus........................ blue 

• rhombus.......................... tan 

• trapezium/trapezoid........red 

• hexagon..................... yellow 

The pieces come in different 

colours, which make it simple 

to identify each. To gain the 

most from pattern blocks, 

students need to be given time 

to explore the pieces that make 

up the set. 

As the students create their 

designs, they will develop an 

intuitive feel for the pieces. 

They will start to notice that 

the side-lengths of each 

piece match (the base of the 

trapezium is twice the length). 

As the students create their 

designs, notice how they 

rotate the pieces to fit snugly 

with other pieces. Refer to the 

shape by name so the students 

start to use the correct 

mathematical names for the 

pieces. Do not be too pedantic: 

if in their haste to create a 

design the students ask for a 

‘green piece’, to begin with, 

that’s OK. 

Students begin to make 

designs naturally. 

1 2 

Make a flower (or tree) using 12 blocks but 

only 2 different shapes. 

3 4 

Make a design with 3 different shapes. 

Repeat the design to start a pattern 



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Make a pet (dog or cat etc.). Draw 

the pet on a piece of paper. 

Now, trace the shape of your pet. Ask a 

classmate to fill it in. 


Developing mathematics 

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Know the blocks 

D T E S 

Introductory Activity 

Purpose 

To formalise knowledge of 

the properties of pattern 

blocks. 

• Make a drawing of each block using the correct colour. 

• Use the correct name. (Discuss other names that may be used; for example a square 

belongs to the quadrilateral family.) 

• How many sides? 

• How many corners? (The correct word is ‘vertices’.) 

1 

List all you know about these blocks. 

Feel the shape(s) 

What shape (or shapes) 

are in the feely-bag? 

3 Make same shapes 

2 

4 

Make pattern block faces 

Make pattern block faces. 



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Guess my shape? 

Discover other ideas 

Once students complete their 

designs, encourage half the 

class to walk around the 

classroom so the students can 

look at other designs. Encourage 

discussion between the designer 

and the observer. 

When students ask and answer 

questions, their understanding 

of the mathematics behind 

the design will improve. By 

eavesdropping on these 

discussions you can gain 

a valuable insight into the 

students’ thinking. Once half the 

class has been on a ‘walk’, ask 

the students to swap roles. 

Diamonds 

While the rhombus shown on 

playing cards is often referred to 

as a ‘diamond’, technically this 

piece should be described as a 

rhombus. 

Discuss a baseball diamond, a 

wedding ring diamond, ‘twinkle 

twinkle little star, like a diamond 

in the sky’. 

CHALLENGE 

This is a trapezium/ 

trapezoid. Why? 

Use 3 or 4 blocks to make a shape. 

Now make the same shape using 

different blocks. 

I have a block in my hand. You have 5 

questions to work out which block. (You can’t 

use colour or shape in your questions.) 

Students need to work out the correct 

shape by asking searching questions. 

Response can be ‘yes’ or ‘no’ only. 

A trapezium is defined as 

having one pair of parallel 

sides. 



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D T E S 

Pattern Block Man 

Purpose 

To create and copy various 

designs using pattern 


Working from the original design and using the same number of blocks at the same time, 

show Pattern Block Man in a variety of poses. 

1 2 

3 

Create Pattern Block Man exactly as shown. 

4 

Pattern Block Man jumps for joy. 

Guide to using this 

photocopiable resource 



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Modelling boards: for ease of 

moving the models, provide the 

students with modelling boards 

of strong cardboard or plywood, 

measuring about 40 x 25 cm. 

A master copy of Pattern Block 

Man is provided for those students 

who need help to make him. 

The master may be copied onto 

an overhead transparency and 

overhead pattern blocks placed 

on to the transparency and shown 

to the students using an overhead 

projector. 

Show another member of the 

Pattern Block Man family. 

Make a pattern block person from 

another planet. 



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Pattern Block Man template 



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D T E S 

Pattern blocks games 

Purpose 

To encourage clear use of 

spatial language to describe 

designs. 

Barrier games 

Barrier games may be played 

with students of all ages. What 

will differ is the sophistication 

of the language used to 

describe how the pieces are 

fitted together. 

Students will use a variety of 

vocabulary to describe the 

position of pieces; for example, 

above, below, in-between, next 

to, opposite, to the left of, to 

the right of, touching, joined 

along an edge … 

Variations 

Have the students sit backto-back. 

Have the builder 

cover his/her design and then 

describe it from memory. 

CHALLENGES 

Estimate, then show: 

• how many triangles will 

cover a flat object 

• how many trapeziums will 

cover a half sheet of A4 

paper 

• the number of hexagons 

needed to cover an 

exercise book 

Pattern blocks are ideal for playing barrier games 

Behind the wall 

This activity is designed to 

be completed by a pair of 

students. A barrier is placed 

between the students (this 

could be a hard cover book or 

similar). One student creates a 

design with a limited number of 

blocks and then describes how 

to create the design to his/her 

partner. The second student 

then tries to recreate the design 

based on the verbal description. 

As students describe their 

designs, listen to the language 

they use. Is it ambiguous? 

Consider the sophistication of 

the descriptions. 

This activity provides some 

useful data that may be used 

for assessment purposes. Look 

for ambiguity or preciseness of 

language and the sophistication 

of the spatial language used. 

Road patterns 

Pattern blocks are ideal for building roads, 

fences and small buildings. 

start 

Extension activity 

finish 

Make a simple design and write a set of 

instructions for recreating that design, either by 

sketching it or by taking a digital photograph. 

Give the written description to another student. 

Other students try to recreate the design 

based on the written instructions. 

The trapezium is on the edge 

of the triangle. 



We recommend that several pairs of 

children attempt this activity at the 

same time. 

Great discussion incentives will be 

obvious. 

Use the road plan (opposite) to 

create patterned roads, following 

the pattern indicated at the start of 

each. 

Note: All pieces except the hexagon 

will fit along the road. 



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start 

finish 



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Please enlarge to 140 % 



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D T E S 

Purpose 

To use the relationship 

between the pattern blocks 

to solve a range of number 

problems. 

How old is Pattern 

Block Man? 

Give the pattern blocks a numerical value 

Gradually increase the complexity of these challenges. 

The challenge of reality 

Once the students understand 

the idea that values may be 

assigned to the pattern blocks, 

a new range of arithmetical 

possibilities emerge. When 

deciding the age of the pattern 

block model all of the four 

operations may be utilised. For 

example: 

If a blue rhombus has a value 

of 4, how many of that shape 

in are in the model? Then 

having ascertained the number 

of blue rhombuses, what is 

the total value (number of 

rhombuses multiplied by 4)? 

Here we need to consider 

the notion of reality in 

mathematics. Although Pattern 

Block Man is quite fanciful, 

the reality emerges from two 

aspects. Firstly, the blocks and 

the associated model are real 

and, secondly, there is a most 

tangible support available to 

encourage the calculation of 

the ages. And for many eightto 

ten-year-old students, that 

reality is sufficient to inspire 

some wonderful arithmetical 

manipulations and counting 

techniques. 

1 

If a triangular shape has a value of 1, what is the 

value of each shape? Why? 

(blue rhombus = 2, hexagon = 6, 

trapezium = 3) 

3 4 

If the trapezium has a value of 6, what is the 

value of each shape? 

(triangle = 2, blue rhombus = 4, 

hexagon = 12, 2 x trapezium = 12) 

Throughout this book, this idea appears 

frequently. On page 28, this concept is 

developed as the students explore a more 

formal approach to fractions. Likewise, 

the same ideas are practised on page 

72 as the students shop for groceries 

using pattern block ‘money’. In all these 

activities, emphasis is focused on ‘what 

the brain says’ rather than expressing the 

ideas with pencil on paper. 

2 

If the blue rhombus has a value of 2, what is the 

value of each shape? Explain. 

(a = 4, b = 4, c = 3) 

If the triangle has a value of 10, what is the value 

of this shape? 

(shape = 120) 



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(a) 

(b) 

(c) 



These challenges are designed for use 

at a Learning Centre where there is a 

collection of pattern blocks and space 

for students to display their models. 

In keeping with the spirit of Learning 

Centres, a challenge may be displayed 

for some time. But frequent, focused 

visits by classroom members will create 

discussion. The background role of the 

teacher will ensure leading questions 

are proposed. The Learning Centre is a 

conversation centre. 

Photograph creations to provide a 

further discussion later. 



8 


How old is Pattern Block Man when the triangle has a value of one year? 

Explain how you worked it out. 

‘Pattern Block Man’ Learning Centre card #1 

How old is Pattern Block Man when the trapezium has a value of six 

years? Explain how you worked it out. 


Pattern Block Man is 

Pattern Block Man is 

Meet the pattern block family 

Pattern Block Man is 40 years old. His wife is 34 years old, his daughter is 16 

and his son is 12 years old. 

Make the Pattern Block Man family when the triangle has a value of two 

years. 


Extended family 

years old. 

years old. 



Display Copy 

Make models of the various relatives of the pattern blocks; cousins, aunty, 

uncle, grandmother, grandfather. Write labels for the ages of each model and 

place them beside the correct ‘person’. 

Challenge: Ask other children and your teacher to calculate their ages. 

R.I.C. Publications ® R.I.C. Publications ® R.I.C. Publications ® R.I.C. Publications ® 




9 


D T E S 

Fences and floors 

Purpose 

To develop the concept of 

perimeter and area. 

Go with the flow 

When we present materials to 

young learners, too often we 

have a set of preconceived 

expectations as to how the 

students will handle them. 

However, the idea of Fences 

came about as the children 

were planning the fields of a 

farm. It was not long before 

paddocks with cattle were 

framed by blue rhombuses and 

the horse field was surrounded 

by hexagons because horses 

jump! The pattern blocks had a 

new purpose. Somewhere amid 

this activity, one of the students 

demonstrated a ‘flat-top’ 

fence by using trapeziums and 

triangles like this: 

Then fences with ‘points’, 

fences with ‘safe’ gaps (that is, 

no animal could escape) and 

fences with carefully designed 

gates were made. A wide 

variety of fences was created. 

One day, one of the students 

arrived with a small collection 

of zoo animals. Proudly, he 

announced ‘I am going to 

make fences for each of these 

animals so that none will 

escape’. 

All this was unplanned by the 

classroom teacher, who showed 

great skill as she encouraged 

pattern making via the fences. 

In contrast, Floors was inspired 

by the elementary science 

study, Teachers guide for 

pattern blocks (1970). 

1 

Fences 

Provide the students with a 

variety of toy animals and 

ask them to create fences 

to keep them in. 

2 Floors 

An introduction to perimeter and area 

‘Take some blocks and cover up a flat space. Make a “floor”.’ 

This will lead to a flurry of new creations; somehow 

the students just have to make patterns that cover a 

flat space. 

As a result a ‘Floor exhibition’ can be created, not 

unlike tiling patterns which adorn ceramic retail 

shops. 

With no leadership from the teachers, the students 

will set limits/conditions on the design of the floors. 

Here are some different conditions: 



Display Copy 

• Make a floor using two different shapes only. 

• Use and and one other shape to 

cover the floor space. 

• Make a design in the middle of the floor and 

repeat it to completely cover the floor. 

• Design a floor which has one continuous colour 

from one side to the other. 

These challenges could be used as Learning Centre 

challenges. 



10 


Expanding the shape 

experience 

Shapes can vary in size and change orientation 

A stage in understanding 

To many young students 

is a triangle, 

is an ‘upside down’ triangle, 

and frequently 

is not a triangle at all. 

When it comes to recognising 

other three-sided shapes, much 

experience is needed to state 

confidently that a three-sided 

shape is a triangle and that there is 

a range of distinct triangles in the 

shape group. 

1 On the overhead projector, display a 2 Encourage students to create and project 

design on the wall using transparent their own designs and patterns. As an 

pattern blocks. Instruct the students to active Learning Centre, the overhead 

use pattern blocks to copy that design, projector can provide an important 

then ask a classmate to check it. 

focus for developing worthwhile shape 

experiences. 

• Show another design on the ceiling. 

3 

• Focus a design on a corner in the 

classroom. 

• Alter the alignment of the overhead 

projector so the design loses 

accurate proportion. 

All these challenges are aimed 

at providing a shape recognition 

experience. 

Some classrooms use 

a lightbox with the 

transparent pattern 




Display Copy 

D T E S 

Purpose 

For students to understand 

that a shape is still the 

same shape regardless of 

orientation or size. 

Making shape real 

A colleague taught 7-year-olds in 

the primary school. One day she 

observed that a large number of 

the students in her class were 

having difficulty recognising 

various shapes, other than in a 

close-up situation or those that 

were aligned as they so often 

are in posters and textbook 

diagrams. 

It was obvious those students 

were struggling with the concept 

that a shape is a shape no 

matter its position or size. 

This shortcoming was tackled in 

a variety of ways: 

1. Using transparent pattern 

blocks and the overhead 

projector to create a variety of 

shape pictures. 

2. (a) Students identified various 

shapes in a feely-bag. 

(b) Shapes were placed in 

clenched hands and the 

student challenged to 

identify the shape. 

(c) Shapes were 

photographed from various 

angles and printed onto 

cards. Students were 

challenged to sort the 

cards into shape groups. 

(d) Solid pattern blocks were 

placed on the overhead 

projector to form an 

opaque shape. Students 

were challenged to 

recreate that shape on 

their tables using the 

pattern blocks. 

Frequently, the conservation of 

shape is overlooked as space 

activities are presented to 

students. 



11 


D T E S 

Picture puzzles 

Purpose 

To match blocks to figures 

(congruency). 

To cover figures with a given 

set of blocks. 

Option 1 is the simplest as 

it simply involves students in 

matching individual pattern 

blocks to fit into the design. 

Option 2 is more sophisticated as 

different pattern blocks may be 

used to complete the design. For 

example, instead of using a single 

yellow hexagon a student might 

use two red trapeziums or six 

green triangles to cover the same 

area. Challenges such as ‘fill the 

design using the least/most blocks’ 

can be set. 

1 

2 

Using pattern blocks to build shape recognition skills 

Create a design to fit onto a single sheet of paper. We have created a design to illustrate 

each activity. 

The students will require some blank white paper and some pencils. Ask the students to 

create a design that fits onto a single blank page. 

Depending on the age and ability of the students, several options exist. 

Encourage students to create a set of similar challenges. 

Trace around each individual shape in the design. 



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Trace around the outside of the design. 

3 

Option 3 is even more difficult 

as the size of the design and the 

Pattern Blocks do not match. The 

original drawing and the final 

version will be the same shape (not 

the same size) but not congruent 

(same shape and size). 

Some teachers prefer to introduce an 

‘in-between’ stage by indicating the 

shapes in the reduced model. 

Trace around the outside of the design and reduce or enlarge the design on the 

photocopier. Then recreate the original design. 



12 


Fill the shapes 

D T E S 

The obvious 

shape is 

not always 

placed on 

the board, 

necessitating 

even more 

strategic 

thinking. 

The players 

have no 

guidance 

as to which 

blocks to 

use. 

Pattern block challenges involving strategy 

Fill the shapes game #1 


Game 1 

Game 2 



Game 1 

Two players take 

turns to choose one 

of these shapes to 

place on the design. 

The last player to lay 

down a block scores 

a point. 

Game 2 

Up to four players, 

in turn, may place a 

pattern block on the 

design. Last to play 

scores a point! This 

is an ideal challenge 

for a team game: two 

groups of students 

planning the most 

strategic moves. 



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Game 3 

Use any pattern block 

shapes to fill in these 

shapes. 

Purpose 

To develop strategic thinking 

while fitting shapes to a 

particular area. 

Developing confident 

shape identification 

As a part of the development of 

spatial skills in young students, 

we realised that the use of 

pattern blocks strengthened 

an understanding of formal 

shape; ‘formal’ as defined by the 

requirements of the mathematics 

curriculum. Consequently, we 

followed a sequential approach, 

using DTES as the underlying 

principle. 

D Students explored the pattern 

blocks, even inventing their 

own names for the pieces. Of 

course, the colours helped; 

many students identified the 

blocks by their colour. And, 

naturally, the square was 

recognised by most! 

T Throughout this process 

students chatted about the 

different blocks, agreed on the 

various characteristics of the 

pattern blocks and gradually 

developed a common 

communication facility. 

E At the same time, the students 

were drawing (with the aid of 

the pattern block template), 

and developing activities to 

challenge their classmates 

S Simultaneously, the formal 

names of the shapes were 

introduced. We emphasise 

that formal names for the 

shapes were not featured 

in the early stages of this 

process. 

As a result of this experiential 

sequence, students identified the 

various blocks with confidence. 



13 


Game 1 



Game 2 



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14 


Game 3 








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Enlarge by 130% to be the correct size for the pattern blocks. 



15 


1 

6 

Fractions 

1 

3 

5 

6 



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1 

3 

1 

2

We appreciate that for 

many students (and some 

teachers) fractions can 

be a very difficult topic. In 

many cases it is because 

students have been rushed 

through to the symbolic 

stage of fraction work. For 

example, many adults might 

remember when dividing 

one fraction by another you 

turn one upside-down and 

multiply. Which one should 

you turn upside-down? Does 

it matter? Why? 

In this section we lay the 

foundation for understanding 

fractions using pattern 

blocks as a model. There 

are several models that may 

be used to illustrate fraction 

relationships. Students should 

experience a variety of models. 

When using pattern blocks, 

the fraction relationships are 

based on the area of four key 

pieces: the yellow hexagon, red 

trapezium, blue rhombus and 

green triangle. 

Common (vulgar/proper) 

fractions 

Percentage 

Ratio 

Some of the ways we can 

see fractions 

Whole to whole 

Fractions: one picture 

For example, when a number is 

compared to another number; such 

as 5 is half of 10. 

shown as the numerator 

‘over’ the denominator, where 

the numerator is less than the 

denominator. 

expressed as %, linked to 100 (X 

out of/in every 100). A very special 

fraction with a denominator of 100. 

Involves a comparison of two 

quantities that may be expressed as 

a fraction. For example, when making 

cordial we might mix one part cordial 

with five parts water. The ratio would 

be 1:5 but altogether there are six equal 

parts therefore 1 / 6 

of the mix would be 

cordial and 5 / 6 

water. Note the ratio 1:5 

is not the same as 5:1, that would be 

undrinkable! 

common/ 

vulgar/proper 

fractions 

improper 

fraction 



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Part to whole 

The most common use of fractions. 

Note: Portions need to be equal. 

mixed 

number 

Describe a subset 

within a group of 

objects 

Three-quarters of the marbles are 

made of glass. 

equivalent fractions 

Depict division 

Fractions are 

rational numbers 

1 

/ 2 

may be considered as 1 ÷ 2 

That is, you can count by fractions. 

decimal 

fraction 0.5 

17

D T E S 

Picture values 

Further challenges 

Design 1 

• If the triangle = 2, what is 

the value of the design? 

Design 2 

• This design is worth 48. 

What is the triangle worth? 

• Cover the design with more 

than 20 blocks. 

• Cover the design with the 

fewest number of blocks. 

• Work at the E developmental 

stage: that is, explain as 

best you can how you 

achieved the solutions. 

On your own 

Purpose 

To discover the relationship 

between the area of various 


• Revise the procedure: Create 

a design worth 19 when the 

triangle is worth 1. 

• When the triangle is worth 3, 

make a series to show 3, 6, 

9, 12 etc. 

• When the hexagon is worth 

3, write the value of each 

block. 

Using these blocks only, if a 

Picture values design #1 

= 1, what is the design worth? 

Working at the DTE stages of development, the students will 

delight you with their numerical dexterity. When asking for an 

explanation do not, at this stage, expect or demand symbolic 

notation. 

Students might respond: 



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• The design is worth 29 because I put 29 triangles on it. 

Guide to 

using this 

photocopiable 

resource 

Design 1 

If the green triangle is 

given a value of 1, work 

out the value of the 

design. Compare results. 

Some students will 

cover the whole design 

with green triangles, 

whereas others may be 

aware that six green 

triangles fit over the 

yellow hexagon, three 

over the red trapezium 

and two over the blue 

rhombus. 

Restrictions may be 

placed on the blocks 

used to make the 

design; for example, 

the students can use 

only 15 blocks in the 

design. This will help 

avoid students making 

grandiose designs that 

take up most of the 

lesson time with building 

rather than working 

mathematically to solve 

the problem. 

Design 2 

The design is worth 

24. What is the triangle 

worth? Explain. 

Give the blue rhombus 

a value if the design is 

worth 48. 

Note 

Pattern Block Man (see page 4) 

is an ideal model for this series 

of activities. 


• I didn’t do it that way. I found that I could use different blocks and 

I gave them each a value. Like the hexagon is worth 6. 


18 

Encourage students to share their ideas. 







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19 


D T E S 

Cover-ups 

Purpose 

To develop problem-solving 

strategies. 

To develop fraction 

awareness. 

Developing a model of 

fractions 

These activities will firmly 

establish the whole/base 

concept, so important in the 

understanding of fractions. Even 

though a sound appreciation 

of area is essential, many 

developing ideas will be 

reinforced. For example, the 

student will know already that 

two trapeziums will completely 

cover a hexagon. 

We recommend the students 

‘build’ a cover-up reference 

board by cutting out the various 

shapes and aligning them in the 

same manner as the pattern 

blocks. Encourage a student 

to create a ‘display’ which is 

meaningful to him or her. 

We are working at the DTE 

stages of development. 

1 

2 

Cover-ups 

Developing sound fraction concepts 

Students, I can cover the hexagon with two trapeziums. (This could be demonstrated on 

the overhead projector.) 

How many different ways can we cover the hexagon? 

How will we know we have found all of the possibilities? 

Ask the students to consider these cover-ups. 

Devise a code 

I want you to describe one of your models but you cannot 

photograph, draw or talk about your description. 

Why are they the same? 

Why are they not the same? 

Guide to using 

this photocopiable 

resource 



Display Copy 

Important note: 

For these activities, do 

not use the square or the 

tan rhombus. Why? Those 

shapes do not naturally 

cover the other shapes 

and the surface area 

relationship is not easy to 

recognise. 

We are encouraging students to think of fractions as an 

alternative recording method. 

Use the shapes on the 


to record various ideas. 

Colour, cut and paste 

these shapes to create a 

collection of cover-ups. 



20 




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21 


D T E S 

Purpose 

To develop a deep 

understanding of the symbolic 

representation of fractions, 

especially halves, thirds and 

sixths. 

Fractions – 1 

Developing an understanding of the written form (symbol) 

A wide range of 

concepts 

Working the cover-up ideas 

with pattern blocks requires 

a fundamental understanding 

of the concept of area; the 

relationship between the various 

blocks is based on area. While 

the concept of area is strong in 

young students, their ability to 

both verbalise and operate in 

these notions is limited. Hence, 

the students often develop 

an understanding of the term 

‘cover-up’ early because it is 

frequently used in their lives. 

We recommend students talk 

and write their ideas from 

their level of understanding. A 

student’s struggle with the idea 

of fractions may arise from 

a number of causes. As the 

students discuss their fraction 

ideas and write about them, 

misconceptions will become 

apparent. 

While the students enjoyed 

these cover-up activities, we 

found the strengthening of 

ideas to be an ongoing process, 

rather than a short series 

of lessons/activities. Young 

students developed a wonderful 

confidence in handling fractions. 

1 

2 

Study this model carefully. What do we observe? 

Among the student’s responses you will hear 

• Three blue rhombuses will cover one yellow hexagon exactly. 

• You can divide a yellow hexagon into three equal pieces and one of the equal pieces will be a blue 

rhombus. 

• One makes three. 

When asked to explain the last comment ‘One makes three’ more clearly, many students will 

explain that three of the same (equal) parts can make one larger piece (whole). 

Using the yellow hexagon as the base, describe the blue rhombus. 

Various answers may include a variety of reactions. 

one out of three 

one out of six 

make 



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three out of six. 

Continue to develop these ideas. Allow students to explore ideas but please do not force 

formal (adult) fraction concepts on them at this time. 



22 


3 Developing the S (symbolic) stage 

While many concepts are involved in this process, we have found this verbal/symbolic 

approach to be very successful. With experience, a more formal and conventional 

understanding will evolve. 

Teacher: I will take one blue rhombus off the yellow hexagon base. 

(On paper or the board write ‘1’ as you say this): 

I have one equal part. 

Students: You have one out of three equal parts. 

Teacher: How many equal parts in the base? 

Students: Three 

Teacher: (Write 3 on the board as you say this): There are three equal parts in the whole 

(pointing to 3). I have one out of three (pointing to 1) equal parts. How can I write 

this so you know what I am talking about? 

Take suggestions and develop an agreement on the generally accepted form of writing 

this relationship. At this stage, the students will understand the formal presentation in 

this manner. 

1 

3 

This is a huge step in formal understanding. 

Now consider: 

I have one out of 

three equal parts. I 

can write this as 

1 

3 


two equal parts. I 


I have one part 

out of 

a base of three (whole) 

There are two out 

of three equal parts 

remaining. I write this as 

2 

3 

There is one out of two 

equal parts remaining. 

I can write this as 

A wide-ranging discussion 

Note: When using pattern blocks the 

fraction family halves, thirds and 

sixths will be emphasised. 

Questions need to be structured in 

order to bring out the relationship 

between these fractions. 

Students will need to be exposed 

to different models such as fraction 

strips in order to experience other 

fraction families such as halves, 

fourths/quarters and eighths. 



Display Copy 

1 

2 

1 

2 


six equal parts. I 


1 

6 

There are five out of six 

equal parts remaining. 

I can write this as 

5 

6 



23 


D T E S 

Fractions – 2 

Purpose 

To develop an understanding 

of equivalent fractions. 

To complete addition of 

fractions with like (or closely 

related) denominators. 

Understanding symbols 

Take time and care as the 

symbols are introduced and 

reinforced. 

+ 

– 

x 

÷ 

= 

With experience, students will 

realise: 

• Subtraction reverses 

addition and addition 

reverses subtraction. (+ –) 

• Division is the successive 

subtraction of equal groups 

and multiplication is the 

successive addition of equal 

groups. ( x ÷) 

• Multiplication is the inverse 

of division and division is the 

inverse of multiplication. 

(x ÷) 

• Equivalence does not mean 

exactly the same, rather 

equal in value, measurement 

or effect. We believe young 

students can comprehend 

the subtle differences 

between the two terms. 

1 

2 

The shaded part is called 1 6 . Why? 

Expect an answer like ‘Because we are 

showing one out of six equal parts’. 

Understanding the written form 

is equivalent to 

Two-sixths is equivalent to one-third. 

2 

6 is another name for 1 3 


Three-sixths is equivalent to one half. 

3 

6 is another name for 1 2 

Using the cover-up models, show all the 

different ways to make 5 . For example; 

6 

1 

6 + 1 6 + 1 6 + 1 6 + 1 6 = 5 6 

OR 

Guide to using these 

photocopiable resources 

Fraction symbols 

(p. 25) 

• Cut out each fraction symbol 

to make a flashcard. 

• Choose a flashcard. Students 

show that fraction using 


• Display a fraction flashcard at 

a Learning Centre. A display 

of all the models of that 

fraction is created. 

• Use all the fraction flashcards: 

create pattern blocks models 

to match each. 

Creating fraction sentences – 1 

(p. 26) 

• Cut and colour the fraction 

shapes from page 21. 

Glue them onto the blank 

hexagons to illustrate 

different fractions. 

• Students explore all the 

different ways a fraction may 

be created. 


(p. 27) 

Use this photocopiable resource 

to reinforce different ways of 

‘seeing’ fraction relationships. 

Ensure the students can explain 

each statement, especially 

the less conventional fraction 

sentences. 

You might say this is a write-talk 

sheet! 

(Note: While we are working with 

1 

/ 2 

, 1 / 3 

and 1 / 6 

, the knowledge 

gained by the students from these 

activities will transfer to other 

fraction relationship patterns; for 

example 1 / 2 

, 1 / 4 

and 1 / 8 

). 



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1 (whole) – 1 6 = 5 6 

1 

6 + 1 6 + 1 2 = 5 6 

OR 

1 – 1 6 = 1 6 + 1 6 + 1 2 

Discover all the different arrangements. 



24 


1 

2 

Fraction flashcards 

1 

6 

4 

R.I.C. Publications ® R.I.C. Publications ® 







1 

3 


2 

6 

5 

2 

3 


3 

6 

Fraction flashcards Fraction flashcards Fraction flashcards 



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1 


6 

6 

2 

2 

OR 

3 

3 

OR 

6 

6 

Fraction flashcards Fraction flashcards Fraction flashcards 



25 


Fraction key 


1 

6 

1 

6 

1 

6 

1 

2 

1 

6 

1 

6 

1 

3 

1 

3 

1 

6 

1 

1 3 

1 2 

3 1 

3 

1 

3 3 

3 

1 

2 

1 

2 

2 

2 

1. Show, then write a fraction sentence of, all the different ways to make 5 / 6 

. Here is an 

example. 

5 

6 = = 

5 

6 = 

5 

6 = = 

= 

5 

6 = 

= 

1 

3 

1 

+ + 

6 



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1 

3 

= 

5 

6 



26 



Fraction key 

1 

6 

1 

6 

1 

6 

1 

2 

1 

6 

1 

6 

1 

3 

1 

3 

1 

6 

1 

1 3 

1 2 

3 1 

3 

1 

3 3 

3 

1 

2 

1 

2 

2 

2 

1 

2 

1 

3 

1 

6 

2 

3 

1. Write fraction sentences to match each pattern block model. Here is an example. 

2 lots of 1 make 1 whole 

• 

2 

take 1 

• 

2 from 1 whole: 1 2 

(1– 1 

• 2 = 1 2 ) 

There are more you can discover. 

• 

• 

• 

• 

• 

• 

• 

• 

• 

is left 



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5 

6 

• 

• 

• 



27 


D T E S 

Purpose 

To explore the relationship 

between pattern blocks and 

use this to generate thinking 

in number. 

An understanding of 

appropriate language is 

essential 

The development of these 

fractional ideas is a multifaceted 

function for most students. 

Full understanding depends 

on a student appreciating 

many fundamental concepts; 

for example, we talk about 

relationships but are we (the 

teachers) or the students aware 

of the meanings being used? 

Many dictionaries list these 

words: relate (relating, 

relater), related, relation, 

relational database and 

relationship. Each of these 

words has a definite meaning 

or implication. In our work 

with students at all levels of 

primary school, we discovered 

it was essential to develop 

vocabulary understanding 

and, with that, appreciation, 

before we felt confident that 

a mathematical concept had 

been fully understood. In fact, 

we developed the maxim ‘A 

great deal of mathematical 

failure is really a vocabulary 

failure’. This realisation shaped 

the subsequent presentation of 

mathematical ideas, especially 

fractions, to the students. 

Important note: 

We created confusion 

when we announced that 

the hexagon had a value 

of 6 as some students 

counted the sides, thereby 

losing the area relationship 

of the blocks. 

1 

2 

3 

4 

Giving the shapes 

value 

Simply, in order for this activity to work, students need to have developed a concept of the 

relationship between certain surface areas. 

Start the idea by showing the triangle. 

If the triangle has a value of 1, what is the value of the blue rhombus? (answer: 2) 

Turn the question around. 

If the blue rhombus has a value of 2, what is the value of the triangle: (answer: 1) 

Continue. 

If the triangle has a value of 1, what is the value of: 

the trapezium? (answer: 3) 

the hexagon? (answer: 6) 

two blue rhombuses? (answer: 4) 

a trapezium and a triangle? (answer: 4) 

Experiment with more challenges. 

• If the triangle is worth 1, make a flower 

worth 12. 

Compare the various solutions. 

• If the blue rhombus is worth 3, create a 

design with the value of 27 using at least 

two different blocks. 



Learning centre challenges 

(pages 29–31) 

These cards may be laminated 

and used as thought starters 

at appropriate Learning Centre 

stations or they may be used to 

inspire the teacher to present a 

series of verbal challenges. 

Encourage students to create 

challenges. From the quality of 

the activities they devise you 

can assess the depth of their 

understanding. 

Once the students have a good 

knowledge of the relationships 

between the pattern blocks, 

challenges of this type may be 

used frequently to strengthen 

mental mathematics proficiency. 

Incidentally, when using these 

activities we discovered our 

own mental mathematics facility 

improved out of sight! 



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28 


= 2 

Shape value card #1 

= 12 


= 6 


The hexagon has a value of 2 

1. Make a model worth 9. Use different shapes. 

2. I have 6 hexagons. What is the total value? 

3. In my hand, I have a two-coloured collection worth 6. 

What blocks might I have in my hand? 

4. To make a total of 12, I have used 6 blocks. 

Which blocks did I use? 


1. Name the values of the blue rhombus, triangle and trapezium. 

2. Which block is equal to one-half of 12? 

3. Find the blocks which show 6, 12, 18, 24, 30 and 36. Explain 

why. Discuss the pattern. 

4. How many blue rhombuses do I need to show 20? 


1. Make a model worth 11. Use three different blocks. 

2. Make many different models of 11. 

3. Make a model of 7: you cannot use the hexagon. 

4. Find the value of 5 blue rhombuses and a trapezium. 



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= 3 



1. Name the values of the blue rhombus, triangle and trapezium. 

2. Which 3 blocks are equal to one-half of 10? 

3. What is the value of 11 triangles? Make the same value with 

the smallest number of blocks. 

4. How many blue rhombuses do I need to show 7? 




29 


= 1 


= 6 


= 1 



1. Make a model worth 9. Use different shapes. 

2. I have 6 hexagons. What is the total value? 

3. In my hand, I have a two-coloured collection worth 6. 

What blocks might I have in my hand? 

4. To make a total of 6, I have used 6 blocks. 

Which blocks did I use? 


1. Make a house worth 40. Write the values used as an 

equation; for example, (6 x 6) + 3 + 1 = 40. 

2. Make a model using blocks with these values: 

3 x 2, 5 x 6, 7 x 1, 2 x 3. 

3. Build a fence with 5 panels: each panel is made up of 

1 x 6, 2 x 2 and 2 x 1. 


1. (a) Write the value of the blue rhombus, triangle and trapezium. 

(b) Write the value of 3 blue rhombuses. 

(c) Find the value of 12 triangles. 

2. Write a statement to describe 3 hexagons, 2 trapeziums and 

7 triangles. 

3. Find 2 blocks to equal 5 triangles and 2 blue rhombuses. Write 

a simple statement to explain this collection. 



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= 3 


1. (a) Write the values of the trapezium, blue rhombus and 

triangle. 

(b) What is the value of 13 triangles? 

2. My collection has 3 hexagons, 3 trapeziums and 3 blue 

rhombuses. What is the collection worth? Write a statement to 

show these facts. 





30 


= 1 


= 1 2 


= 1 3 


The rhombus has a value of 1 

1. Name the values of the hexagon, triangle and trapezium. 

2. Which block is equal to one-half of 1? Why? 

3. Find the blocks which show 3, 6, 9, 12 and 15. Give reasons 

for your decisions. 

4. How many triangles are needed to show 13? 

The trapezium has a value of 1 2 

1. What is the value of 15 triangles and 4 blue rhombuses? 

2. I have 6 hexagons. From that collection, I take 2 trapeziums, 

5 blue rhombuses and 2 triangles. What is the value of the 

remaining collection? 

3. To make a total of 5, I have used 13 blocks. Which blocks did I 

use? Create 3 more problems like this. 

The triangle has a value of 1 3 

1. Show 1 2 3 and 3 1 3 . 

2. Using the least blocks possible, show 9 x 1 3 . Explain. 

3. Make a model of 5 1 ; you cannot use a trapezium or a triangle. 

3 

4. Find the value of 5 hexagons. Explain your answer. 



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5. Draw the blocks you use to show 2 2 3 + 3 1 3 . 




A trapezium is to a hexagon as a triangle is to a rhombus 

1. A rhombus is to a hexagon as a triangle is to a …? 

2. A triangle is to a hexagon as a trapezium is to …? 

3. True or false? A trapezium is to 4 hexagons as a triangle is to a hexagon 

plus a blue rhombus. Explain. 

4. A triangle is to a trapezium as a rhombus is to ...? 





31 


D T E S 

Purpose 

To work with fractions in a 

non-algorithmic fashion. 

To discover relationships 

between combinations. 

Some suggestions 

Often, we consider fractions 

in isolation from the rest of 

the number system, when in 

fact fractions are an integral 

part of that number system. 

Just as with whole numbers, 

a conceptual understanding 

of operations, magnitude 

relationships and number 

is essential. The activities 

presented here require 

multifaceted understandings to 

complete. Initially, it is essential 

that the appropriate pattern 

blocks are available for use 

by the students: in time, they 

will be able to perform these 

calculations mentally. 

Of course, there may be a 

variety of pathways to finding 

the correct solution. Make sure 

good discussion clarifies the 

thinking of the students. As this 

verbal skill develops (through 

DTE), students will begin to 

translate their ideas into, firstly, 

a written form (E), then the 

more mature symbolic form (S) 

will emerge. These activities 

initialise the complete DTES 

developmental process. 

1 

2 

Pattern block 

fractions 

Expanding the base to really ‘know’ fractions 

We have discovered there are interesting relationships between the surface areas of certain 

pattern blocks. For example, we know that 2 triangles will cover the blue rhombus exactly. 

Consider 

If has a value of 1, we know that has a value of 1 2 . 

So the value of will be 1 2 of 1 2 which is 1 4 . 

With that knowledge, discover the value of this shape. ( 3 4 ) 

Use the same pattern thinking to solve these relationship groups. 

I have a trapezium and a blue rhombus. I give 

this group the value of 1. What is the value of the 

triangle? Why? 

(triangle = 1 5 ) 

I have a shape formed by a trapezium and a 

triangle. Close your eyes. See the shape. How 

many triangles in that shape? (4). Keep your eyes 

closed: tell the value of the blue rhombus. ( 1 2 ) 





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This shape has a value of 1. 

Students enjoy completing these 

tasks. 

Use pattern blocks to show how 

the answer was reached; for 

example: 

• What is the value of the triangle? ( 1 9 ) 

• Why does the blue rhombus have a value of 2 9 ? 

+ = 1 / 4 

, then 1 is 

2 • Is it possible to show 

3 

(Yes, the hexagon) 

with a block? Why? 


32 

Invent more of these challenges and 

present them frequently. These are far more 

beneficial to the development of number 

facts than reciting tables. 

Capitalising on their enthusiasm, 

challenge them to create more 

tasks. 



Pattern block fractions 

If + = 1 3 , what is 1? If + = 1 , what is 1? 

3 

Pattern block fractions card #1 Pattern block fractions card #2 

If + = 1 4 , what is 1 1 2 ? If + = 1 2 , what is 1 2 3 ? 




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If + = 2 3 , what is 2 9 ? If + = 3 4 , what is 1 1 2 ? 




33 


Space, 

shape 

120º 120º 

120º 120º 

120º 120º 

and spatial 

relationships 



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Teachers need to constantly 

value and use visual imagery 

as part of their teaching 

repertoire with the view to 

developing and enhancing the 

ability of their students to use 

visual imagery. 

Dept of Education and the Arts Tasmania (1994) 

An overview of the space strand: Mathematics K–8 

Guideline: p.7 

1 

2 

3 

4

Shape and space is the less formal name for geometry, which literally means 

earth measure. Rather than learn a body of facts or shape names the emphasis 

in this section is on the development of spatial ideas. As students explore these 

ideas they will develop the language associated with shape and space. 

We use spatial ideas for a wide variety of practical tasks. We describe 

our surroundings, find our way around and mark out and construct 

living spaces. Spatial ideas are basic to the solution of many design 

problems … 

(A national statement on mathematics for Australian schools, 1991, p. 78) 

Consider these spatial headings (or topics) which may be appropriately 

investigated with pattern blocks: 

Reflections— 

‘maths with a mirror’ 

Symmetry, folding 

New shapes from ‘old’ 

Shapes and enlargements 



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Movements: rotations, flips 

and turns 

Angles 

Tessellation 

35

D T E S 

Purpose 

The students discover which 

pattern blocks may be 

joined together (tessellated) 

to produce larger, similar 

shapes; that is, the same 

shape but not the same size. 

Similarity 

Two figures are similar if they 

are the same shape, but not 

the same size. The size of 

corresponding angles will be 

the same and all sides are in 

proportion. 

Two figures are said to be 

congruent if they are exactly 

the same size and shape. 

Tricky trapezium 

challenges 

• Use four red 

trapeziums to create 

the same shape but 

larger (similar). 

• Make another similar 

trapezium using nine 

red trapeziums. 

Enlarging shapes 

Enlarging by tessellating 

Using the same shape, make the same shape but larger. 

1 

How many triangles are needed for the next 

model in the series? 

2 

Find the same number pattern in these 

series. Why? 



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3 

If you built the seventh model in this series, 

how many tan rhombuses will you need? 

• Explain why the same 

pattern cannot be 

created using the 

hexagon. 



36 


Comparing shapes 

D T E S 

1 

Consider the growth patterns of these squares. 

3 

4 

Purpose 

To discover patterns from 

collection data (pre-algebra). 

2 

3 

1 

The length of a side of the single square is one unit. 

The area of a single square is one square unit. 

It is important that students are given the opportunity to collect data about these square 

arrangements. 

Create this table about the growth pattern of squares. 

Shape Perimeter units Area 

1 4 units 1 square unit 

2 8 units 4 square units 



5 

6 

Students will determine the answers because they have recognised the pattern. 

• Discover any interesting relationships between perimeters 

and area. 

• Is there a definite, constant pattern? 

• Explore a square with a perimeter of 32 units. 

• How many squares (other than a single square unit) in 

squares of 16 square units and 36 square units? 

Now create a table built on a similar pattern but using triangles. 

1 

2 

2 



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3 

4 

DSOP 

The search for pattern seems to 

be a natural function of the brain 

and the real mathematical growth 

occurs when the pattern is tested 

and tested again. When that pattern 

stands up to the formation of an 

hypothesis—that is, it has been 

successfully tested—a sound 

mathematical process has taken 

place. We call the whole process 

DSOP. It works like this: 

1. Collect data: this is happening 

all the time through all the 

senses: sight, touch, hearing, 

taste and smell. 

2. Sort (classify) this data: this 

is a constant function of the 

brain. 

3. Find order in this data. 

4. Now discover a pattern. 

While we are working with the 

understanding of growing shapes, 

other mathematical skills are being 

utilised in the research process. 

Rarely is the cognitive pathway to 

the understanding of a concept the 

same in any individual. 

The concepts of perimeter and 

area have been introduced in this 

activity. See pages 62–65 for a 

series of development activities for 

these ideas. 

Shape Perimeter units Area 

1 3 units 1 triangle unit 

2 6 units 4 triangle units 



5 



37 


D T E S 

Purpose 

To investigate angle sizes of 

the pattern block pieces. 


between the angle sizes of 

various pattern blocks and 

the number of pattern blocks 

shown in the mirrors. 

From experience 

Not only is this series of 

activities therapeutic but it also 

opens a huge range of new 

ideas. In more formal terms, 

students will be exploring 

notions of symmetry, reflections, 

angles and the beginnings 

of thinking about making a 

kaleidoscope. 

We use unbreakable, flexible 

plastic mirrors. To create a 

hinged mirror, two mirrors 

are joined by masking tape 

(because it is more flexible). In 

our classroom there is always 

a ‘Mirror Learning Centre’. We 

have discovered that some 

students, especially restless 

boys, relax at this Learning 

Centre. 

Avoid introducing formal angle 

measurement at this stage. 

Rather, make discoveries like, 

‘I can make 2 triangles’, ‘By 

moving the mirrors I can make 

3 triangles’ … and so on. 

This knowledge will connect 

as we explore angles (pages 

42–44). 

Mirror, mirror 

A simple hinged mirror provides many exciting insights 

1 Exploration 

2 

For the first series of explorations, use the green triangle and a hinged mirror. Explore 

different images as the angle of the mirrors is changed. 

Students will explore designs patiently and quietly for extended periods. (The mirrors are 

cheap enough for each student to have one.) We have observed the students build the 

pattern blocks vertically (stacking one on top of the other). 



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Sorting 

Using the hinged mirror and a green 

triangle nestled in the joining angle of 

the two mirrors, find: 

two triangles 

three triangles 

four triangles 

five triangles. 

Note the angle made by the mirrors 

each time. We do not expect students 

to use formal angle language at this 

point but simply to note whether the 

angle is getting smaller or larger. 

3 

More exploration 

Place the green triangle about 8 cm 

directly in front of the hinge. 



38 


4 

Other shapes 

Experiment with other shapes. 

5 

More than one shape 

Experiment with more than one shape. 



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6 

Assessing students’ understanding 

of the experience 

• Keep a record by photographing the results. 

• Why can you see more blocks when the angle 

of the mirrors is close/tight (acute)? 

• Each time the angle of the mirrors is changed, 

take a photo, making sure the reflections are 

easily seen. When printed, shuffle the photos 

and challenge the students to place them in 

order, beginning with the mirrors aligned in a 

straight line. 



39 


D T E S 

Purpose 

To investigate the angle sizes 

of pattern block pieces. 


between the angle sizes of 

various pattern blocks and 

the number of pattern blocks 

shown in the mirrors. 

Hidden learning 

opportunities 

Recently, we gave a group of 

teachers a collection of pattern 

blocks and an appropriate 

number of hinged mirrors. 

The instruction ‘Explore!’ was 

given. There was almost an 

instant revolt. Calls like, ‘What 

do we have to do?’, ‘What do 

you want?’ were thrown at us. 

We did not respond and, with 

an undertone of grumbles, 

members of the group began to 

look as if they were ‘exploring’. 

Deliberately, we left the room 

for about 15 minutes. On 

returning we were startled by 

the quiet hum of busy attention 

to matters at hand. The group 

members were absorbed by 

their many creations and were 

quietly sharing their discoveries. 

No mention was made of 

appropriate mathematical 

discoveries; rather, many of the 

group were dumbfounded at 

the variety of the designs they 

were creating. One participant 

shared her reaction: ‘At first I 

didn’t know what to do when 

we were sent off to play with 

mirrors and shapes. But after 

I got over the initial shock of 

not being told exactly what 

to do with these materials, I 

began to do some exploring of 

my own and was intrigued by 

the endless possibilities that 

went with creating reflections 

of the shapes’. The next day, 

one member of the group 

returned with a beautifully 

handcrafted kaleidoscope. He 

exclaimed, ‘I appreciate this 

magnificent thing even more 

after yesterday’. 

Create that sense of wonder and 

discovery in your classroom. 


40 

1 

2 

Recording results 

More mirrors 

Immersing in the experience 

Record the angles made by the mirrors to create 3, 5 and 6 images. 

How many squares can you see? Oh yes! you spotted a large 

square made up of 4 smaller squares. 

You know that when two lines meet like this , the space 

between the two lines that join is called an angle. 

There are some special angles, but let us look carefully at the 

square. The space between the joining lines is the same. We call 

the lines ‘right angles’. They are measured by degrees: each angle 

is 90º. 

So, what is the total number of degrees in a square? 

Aha! Three hundred and sixty degrees. 

In the model you have made, how many degrees are there 

altogether? Discuss 

At what angle are the mirrors? 

Discover the angle ideas with a trapezium. Like a square, 

it has four sides, so it has a total of 360º though not all the 

angles are the same. Use the ‘Hinged mirrors exploration 

sheet’ (p. 41). 

Guide to 

using this 

photocopiable 

resource 



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Place the hinged 

mirror with the join at 

the point and open or 

close the mirror on the 

lines. Place a pattern 

block with one corner 

at the point. Explore 

and record your 

discoveries. 



Hinged mirrors exploration sheet 



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41 


D T E S 

Exploring angles 

Purpose 

Using prior knowledge 

such as a right angle is 90º 

or a full circle is 360º to 

determine the angle sizes of 

each of the pattern blocks. 

1 

An introduction to angle 

2 

Angle facts 

An angle is the space between 

two straight lines which meet at 

a common point. The point may 

be called a vertex. 

An angle is measured by the 

amount of turn, using degrees, 

which are indicated by º (finer 

measurement uses the terms 

minutes and seconds). 

Types of angles 

acute angle – less than 90º 

right angle – 90º 

(Note the special sign used 

to indicate a right angle.) 

We have met students searching 

for ‘left’ angles! 

obtuse angle – more than 90º 

but less than 180º 

straight line – 180º 

reflex angle – more than 180º 

but less than 360º 

circle – there are 360º in 

a circle; some may say a 

revolution 

Polygons 

Polygons have a definite number 

of degrees of their internal angle. 

triangle: 180º 

quadrilateral: 360º 

hexagon: 720º 

(Note: Each angle in a regular 

hexagon is 120º) 

Students will discover that all the 

angles of a regular polygon are 

the same. Not all polygons are 

regular—consider the regular 

hexagon in the set of pattern 

blocks with other hexagons. 

3 

5 

I know a corner of a square is a right 

angle. What is the small angle of the tan 

rhombus? 

Use the tan rhombus and the hinged 

mirror to create a picture of twelve tan 

rhombuses. 

We know that the small angle of the tan 

rhombus is 30º. In the mirror we made a 

circle; how many degrees altogether? 

Now I know the smaller angle in the tan 

rhombus, what is the small angle in the 

blue rhombus? 

Combine shapes to make different angles. 

How many degrees in the angle above? 

Use your knowledge to calculate this angle. 

An assessment idea: Using the overhead projector and overhead projector pattern blocks, 

students explain their method for finding angles. 

4 



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42 


6 

7 

Using a 180º protractor 

Introducing a protractor 

The protractor is one type of angle measurer. By using the protractor to accurately measure 

the angle created by the hinged mirrors, we can determine the angles in each block and 

the number of blocks needed to make 360º (the angles in a circle). 

Experiment with the one model as the angles of the mirror are changed. 

Using a 360º protractor 

At what angle does the block completely disappear? 

Try all the pattern blocks. What have you discovered? 

A word about protractors 

Students will experience trouble 

using protractors unless they 

have developed a good sense of 

angle. For example, they should 

be able to classify an angle as 

acute, right or obtuse before 

touching a protractor. Many 

students experience difficulty 

reading the ‘double scale’ on a 

protractor and may read 30º as 

the size of a 150º angle. This will 

not happen if students have a 

good sense of angle first. 

Angle facts 

These ideas will develop as 

students experience these 

activities. 

1 complete turn = four right 

angles = one revolution 

1 right angle = 1 / 2 

a straight 

angle = 1 / 4 

of a revolution 

1 straight angle = 2 right angles 

= 1 / 2 

a revolution 

1 revolution = 4 right angles = 2 

straight angles 

(Thought: When dealing with 

angles, maybe we would be wise 

to call 360º a ‘revolution’ rather 

than a ‘circle’.) 

Relate this angle knowledge to a 

clock; for example: 

1 

/ 4 

past 1 

/ 2 

past 

1 

/ 4 

to ( 3 / 4 

past) o’clock 

20 past 20 to (40 past) 

o’clock 

Students enjoy inventing these 

time signals. Please allow 

students to make the rules; for 

example, the orientation of an 

hour (60 minutes or a complete 

revolution—360º). 





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(see overleaf) 

Exploring angle (p. 44) 

Students will require sound prior knowledge 

before addressing the challenges on this 

photocopiable resource which may be 

considered as an assessment activity. 

Before the students complete the challenges 

ask: 

Which pattern blocks are quadrilaterals? 

How many degrees in each shape? 

How many degrees in a green triangle? 

If the red trapezium has a total of 360º, how 

many degrees in the yellow hexagon? 

Students will be well aware of the attributes 

of each pattern block shape. 



43 


Exploring angles 

Discovery key 

30º 

60º 60º 90º 

1. How many degrees are the marked angles? 

(a) (b) (c) 

(d) (e) (f) 

2. Label the type and size of angle marked in each shape. 

(a) 



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(b) 

(c) 

(d) 

(e) 

(f) 

(g) 



44 


1 

Seen from all 

directions 

The angle of sight changes the view 

Make an interesting model and place it in a clear space. 

Take a photograph of the model from each of the shown positions. Print the 

photographs and, if possible, laminate them. 

2 Photo direction card game 

• Select any card. Place it on the table. 

• Search for and find the next card. You are moving 

clockwise, find the next photograph. 

• Continue until all photographs are aligned. 

The same activity may be conducted using a single 

pattern block as a model. 



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card 1 

card 3 

card 2 

D T E S 

Purpose 

For students to realise that 

shapes remain constant 

regardless of orientation. 

Conservation of shape 

In recent decades, much attention 

has been paid to the concept of 

conservation in the teaching of 

mathematics; especially is this 

notion considered when dealing 

with number and, certainly, 

measurement. But too little 

attention is paid to the development 

of this awareness in relation to 

shape. As an example, students will 

easily recognise a triangle in this 

position: 

When the position is changed, so 

is the students’ perception of the 

shape, hence: 

(a triangle on the side) 

becomes a ‘new’ shape altogether. 

As the young student develops 

shape awareness, the reaction to 

this change: 

to to 

generally elicits the response ‘Aha, 

an upside-down triangle’. 

Interestingly, we discovered that 

when the pattern blocks were 

close by, a young student had 

little difficulty in recognising 

same shapes in various positions. 

Move the same shapes further 

away and we observed hesitancy 

in identifying the shapes. When 

the positions of the shapes were 

changed, recognition problems 

emerged. In summary, students 

need a wide range of spatial 

experience. 



45 


D T E S 

No overlaps 

Purpose 

Students use individual 

pattern block pieces and 

combine them to produce a 

tessellated pattern. 

Discovering which pattern blocks tile (tessellate) 

A tessellation 

connection 

Before introducing the notion 

of tessellation with the pattern 

blocks, thoroughly explore the 

concept of tessellation. Without 

realising it, the students have 

explored tessellating with the 

blocks, but effort needs to be 

directed towards developing the 

key (mathematical) principles of 

tessellating. 

• Study varieties of brickwork 

used in houses and 

pavements. 

• Discover all the applications 

of tiles in a household. What 

happens when a tile does 

not fit exactly in a space? 

Comment on the various 

ways in which patterns are 

generated. 

• Search for tessellating 

patterns which use one 

shape (polygon) only. 

Then, discover tessellating 

patterns which use more 

than one polygon. These 

are called semi-regular 

tessellations. 

Certain shapes may be used to completely cover a surface. For other surfaces, a 

combination of shapes may be needed. Did you know that to completely cover the 

curved roofs of the Sydney Opera House every tile was individually designed? An extreme 

tessellation exercise! 

1 Use nine pattern block squares and 

arrange them carefully to cover an area of 

paper. Make sure there are no gaps. 

2 

We have covered a region so that no part 

is left uncovered. Draw a line around the 

area covered and cut out that shape. 



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46 


3 

Using the pattern block squares, arrange 

them on the square that you cut out so 

that only the vertices touch. (Here is an 

example.) 

4 

Try this. Cover your piece of paper with 

squares arranged like bricks in a wall. 

To keep the pattern, suggest how the 

spaces are filled. 

Why polygons will 

tessellate 

A polygon will tessellate if the 

total of all the angles meeting 

at a common vertex is 360º. 

Here are three pattern block 

examples: 

squares 

90º 

90º 

90º 

90º 

5 

6 

How many squares covered your piece of 

paper? 

Students will chorus that there are gaps 

and you can’t completely cover the paper. 

Each student has a square piece of paper. 

(a) Arrange the paper this way and 

cover it with pattern block 

squares. (This is a spatial exercise, 

some students may have a problem 

as they attempt to align squares in a 

‘vertical’ manner. Discuss ideas with 

these students.) 

Using two pattern block pieces 

The purpose of these challenges is 

to provide a sound setting to develop 

the concept of tessellation: covering a 

surface with no overlaps or gaps between 

shapes. 

(b) Cover the paper using the same shape: 

triangle, rhombus, trapezium, hexagon. 

Collect data by sketching or 

photographing the model and discuss 

findings. 

Cover all of the piece of paper by using two or more pattern blocks. Compare results with 

those of other students. 



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4 x 90º = 360º 

blue rhombus 

60º 

60º 60º 

60º 60º 

60º 

6 x 60º = 360º 

regular hexagon 

120º 

120º 

120º 

3 x 120º = 360º 

equilateral triangle 

60º 60º 60º 

60º 60º 

60º 

6 x 60º = 360º 

trapezium 

60º 60º 60º 

60º 60º 

60º 

6 x 60º = 360º 



47 


D T E S 

Symmetry 

Purpose 

To create shapes using 

pattern blocks that have 

either a single line of 

symmetry or multiple lines of 

symmetry. 

From experience 

As the ideas of symmetry evolve, 

we encourage the introduction 

of a wide range of activities. 

Students’ spatial experience 

depends on two most important 

developmental aspects: firstly, 

physical movement, where the 

internal notions of balance, 

uprightness, level and sameness 

are gradually assimilated; 

and, secondly, the powers of 

observation and the subsequent 

reasoning that evolves. 

Consequently, we recommend 

strongly that the concept of 

symmetry is slowly developed 

rather than thrust upon the 

students as an independent 

subject. 

What is symmetry? 

The word ‘symmetry’ is a noun. 

Therefore, a shape will ‘have 

symmetry.’ A simple definition 

says a shape is symmetrical 

when one half of its shape 

can be placed exactly over the 

other half. This suggests the 

concept of folding is important: 

paint blots, sheets of paper, 

books, newspapers and cutting 

shapes out of folded paper are 

appropriate developmental 

activities. 

We can make a collection of 

lines of symmetry. 

1 

2 

Not all shapes have two identical halves 

Does a face have a line of symmetry? 

Create Pattern Block Man to show one line of symmetry. (A group of students may create 

the model on the overhead projector.) 



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Some objects have more than 

one line of symmetry. 

Explain why only one line of symmetry can be found. If no more than two blocks were 

changed, is it possible to discover more than one line of symmetry? 

Experiment with different arrangements of Pattern Block Man to try to find two lines of 

symmetry. 



48 


3 Students discover the lines of symmetry in the pattern blocks set. 

5 Use three blue rhombuses and three green triangles to 

4 Create models which have two or more lines of symmetry. 

create a number of equilateral triangles (two are illustrated). 

The outline of the shape will be an equilateral triangle which 

has three lines of symmetry. However, because of the 

arrangement of the blocks line symmetry is not preserved. 



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49 


D T E S 

Reflection pictures 

2 

Purpose 

To extend students’ 

understanding of symmetry. 

Developing knowledge 

of symmetry 

These activities are ideal for a 

Learning Centre where students 

experiment freely. Rather than 

insisting on achieving a set 

expectation, allow students 

to develop their personal 

knowledge of symmetry. We 

believe the concepts involved in 

this area of spatial knowledge 

are complex and thus require 

the constant presentation of 

developmental activities. 

1 

The search for lines of symmetry continues 

Using the pattern blocks, make many reflection pictures. Here are some single 

line reflection picture ideas. 

Experiment to create reflection pictures with more than one line of symmetry. 





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These activities are ideal for a 

Learning Centre where students 

experiment freely. 

1. Working as pairs, one student 

builds a shape on one side 

of the line. The other student 

‘matches’—reflects—the same 

shape on the other side of the 

line. 

2. Create models which have two 

or more lines of symmetry. 



50 




‘Reflection pictures’ Learning Centre starter card #1 





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Enlarge to suit your needs. 


51 



D T E S 

Mirror symmetry 

Purpose 

To investigate the line of 

symmetry on individual 


To produce designs with 

single and multiple lines of 

symmetry. 

Developing ideas of 

symmetry 

Students’ ideas about symmetry 

are now strengthening. For 

some students, the idea of 

‘folding’ enables them to ‘see’ 

symmetry in an object. Others 

may prefer to use the idea of 

‘balance’, implying a sameness 

on both sides. 

Sometimes, in an effort to 

make the concept of symmetry 

simple, the folding concept is 

used with regular shapes, such 

as a square. This example does 

show four lines of symmetry 

but little search, thought or 

experimentation is required 

to discover these lines. Not 

only in the conceptual area 

of symmetry but also in 

many shape-related areas of 

mathematics the use of regular 

geometrical shapes can limit 

the development of conceptual 

ideas. In other words, while the 

use of pattern blocks is an ideal 

introduction to symmetry and its 

related ideas, extend the ‘search 

for symmetry’ to a wide range 

of shapes, including the human 

face and cars. Remember, it is 

a sound teaching practice to 

search for symmetrical figures 

and not find them. It is good 

to introduce ideas that do not 

work as long as the students 

understand the reasons why 

these ideas do not work. 

Exploring lines of symmetry with a mirror 

1 2 

To see if an object has a line of symmetry 

place a mirror on a suggested line of 

symmetry. If the whole figure can be 

seen—part object and part reflection—a 

line of symmetry has been found. The use 

of pattern blocks is an ideal medium to 

develop these ideas. 

3 Challenges 

Using a mirror: 

• Make a trapezium look like a hexagon. 

• Arrange a triangle to look like a 

rhombus. 

• Make a square into a rectangle. 

Explain why this is so. 

• Place the mirror on a square to reflect 

a triangle. Has a line of symmetry been 

discovered? (No) Why? 

• Experiment with the red trapezium to 

create: 

a large equilateral triangle 

a longer trapezium 

a small equilateral triangle 

a rhombus 

The cup may be considered to be 

asymmetrical—it has no line of symmetry. 

By placing a mirror, a line of symmetry 

is created between the object and the 

image. 

• Make a rhombus look like a trapezium. 

Has a line of symmetry been created or 

used? Why? 



Display Copy 

A line of symmetry on the original 

shape has not been used to create this 

trapezium, but the new shape has a line 

of symmetry. 



52 


4 Create models to show lines of symmetry using a mirror. 

Note: The first two shapes are symmetrical by shape but not by colour. 

5 

rotate 

Explore transformations on the computer. Most software programs include options to 

rotate, reflect and translate shapes. Encourage students to explore these options. 



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Transformations 

Commonly known as flips, slides 

and turns, often these terms 

are used synonymously with a 

study of symmetry, but care must 

be taken that the concept of 

transformation is understood. 

The reflection (flip), translation 

(slide) and rotation (turn) are all 

examples that leave a shape 

unchanged. 

Shapes may also be transformed 

by enlargement or reduction. 

Flip – reflect over a line 

Slide – 

Turn – 

reflect (flip) 

translate (move) 

square rotated 45º star reflected oval translated 



53 


D T E S 

Round and round 

Purpose 

To explore rotational 

symmetry. 

Rotating shapes to explore symmetry 

Background 

Turning or rotating a shape is 

one form of translation. The 

shape is rotated about a point. 

When examining whether a 

shape possesses rotational 

symmetry, the point is placed in 

the middle of the shape. 

If a shape can be rotated less 

than 360º to fit on itself, then 

it is said to have rotational 

symmetry. If the shape fits on 

itself three times throughout 

a full rotation (360º), then the 

shape is said to have an ‘order 

of rotational symmetry’ of three. 

original 

shape 

rotated 

120º 

shape 

rotated 

240º 

1 

Ask the students to make two copies of each pattern block. Cut one copy of each pattern 

block out (tracing paper is useful). 

Use the overhead projector and a transparency showing the pattern blocks to demonstrate 

rotations. Place an overhead pattern block on top of its associated outline and slowly rotate 

the piece to show how many times the piece fits on top of its outline. 

Ask the students to explore the rotational symmetry of each pattern block piece. Students 

may find it simpler to place the cut out pattern block piece on top of the outline and then 

place a pencil point in the centre of the shape and rotate it. Students will need to record the 

number of times the shape fits over the outline. 

Fits 3 times 

over itself. 

Fits 4 times 

over itself. 

Fits 2 times 

over itself. 



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shape 

rotated 

through 

360º 

Note: 

• If a shape has to be rotated 

360º before it fits over 

itself then it does not have 

rotational symmetry. 

• The rotations above may be 

referred to as a 1 / 3 

or 2 / 3 

turn. 

• You must rotate clockwise. 

Fits 6 times 

over itself. 

Fits 2 times 

over itself. 

No rotational 

symmetry 

Students may notice that the hexagon has six lines of symmetry and also has an order of 

rotational symmetry of 6. Encourage the students to explore this. 



54 


2 

Challenge the students to make shapes with rotational symmetry. 

3 

4 

Many shapes have both line and rotational symmetry. Challenge the students to create a 

shape that has rotational symmetry but NOT line symmetry. 

Look at the design of hubcaps on alloy wheels. Take some digital photos, print them, mark in any lines of symmetry 

and note the order of rotational symmetry. 



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CHALLENGE 

Make a series of shapes 

with pattern blocks that have 

orders of rotational symmetry 

from 2 to 8. 



55 


D T E S 

Purpose 

To work mathematically by 

applying problem-solving 

skills, such as working 

systematically, to produce all 

possible combinations. 

To recognise rotations or 

reflections of shapes. 

Background 

The development of spatial skills 

is most essential and plays an 

important role in most aspects of 

mathematics. 

At first glance, this activity may 

seem trivial, however … 

A good spatial sense is a basic 

indispensable capability that all 

individuals should possess … 

An important component of this 

indispensable spatial capability 

is the ability to perceive 

and ‘hold’ an appropriate 

mental image of an object or 

arrangement. 

Department of Education and 

the Arts Tasmania (1994) An 

overview of the space strand: 

Mathematics guidelines K-8 p. 5 

1 

New shapes from old 

Joining shapes 

3 Combine the red trapezium and the blue 

rhombus. 

Creating shapes by combining other shapes 

Two green triangles may be joined along 

an edge to produce a new shape (in this 

case a rhombus). 

Note that shapes need to be joined along 

the entire edge, not like this: 

Are these the same shapes? 

Yes! Orientation of shape does not 

change a shape, even though it may 

appear different. 

Investigate all the shapes that may be 

made by joining three triangles. (one) 

2 By combining the red trapezium and 

the green triangle, create many different 

shapes. 

large equilateral triangle 

parallelogram 

irregular hexagon 

4 Use three or more shapes to create a variety of 

hexagons. 



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another trapezium 

irregular pentagon 

5 ‘Six sides plus three sides makes five sides’. 

Use a hexagon and a triangle to prove this 

statement makes sense. 

Discover other interesting combinations 

where more sides when combined make 

fewer sides. 

6 

There are three different figures that may be made using 

four triangles. 

Use four pattern blocks of the same shape to create 

new figures. Operate with the same rules each time. 



56 


Hexiamonds 

D T E S 

Shapes with equilateral triangles 

A hexiamond is a shape made up 

of six equilateral triangles. Each 

must join along a full side. 

1 Make all the hexiamonds. How do you know you have discovered them all? (There are 

twelve hexiamonds.) Caution: Beware of reflections. 



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Purpose 

To work mathematically 

to find all possible 

shapes made from six 

equilateral triangles. 

A shape is a shape 

is a shape 

I was challenging a young lad 

about making as many shapes 

as possible. Rather carelessly I 

proposed, ‘How many different 

shapes can you make with two 

squares?’ 

Picking up two square pattern 

blocks the young lad began 

making shapes. His concentration 

was evidenced by a prominent 

salivating tongue. ‘One shape, 

two shapes …’ he counted as he 

made a series of different shapes. 

One shape 

Two shapes 

‘Three shapes, four … Blow this!’ 

he shouted suddenly. (In fact, 

for a five-year-old his expletive, 

although unprintable, was most 

colourful.) ‘I’m not doing this. It is 

infinity.’ 

I learnt two most important 

lessons from this experience. 

Firstly, he was following 

instructions carefully and 

correctly; and secondly his ‘inside 

the head’ knowledge was far 

greater than I ever dreamed. 

2 

Create a flow chart to demonstrate how 

all the hexiamonds were discovered. 

3 

Using heavy paper or light cardboard, 

cut out all the hexiamond shapes and 

use them to create a square, a large 

equilateral triangle and a parallelogram. 

Students may create challenging 

puzzles/games to introduce to students 

in other classrooms. 


57 


with pattern blocks

D T E S 

Purpose 

Students rotate and slide 

pattern block pieces to fill a 

region. 

Students work mathematically 

to fill a region according to 

specific criteria. 

Making triangles 

A wide range of equilateral triangles may be created 

using a variety of pattern blocks 

You will need 

Method 

Triangle fill (a barrier game) 

pattern blocks a barrier copies of p. 59 

• Two students face each other with the barrier hiding their triangles. 

• Student 1 fills his/her triangle with pattern block shapes. 

• Student 2 asks questions to determine which pattern blocks were used to 

fill the triangle. The first student may only answer yes or no. 

• Student 2 keeps asking questions until he/she is satisfied the other triangle 

has been replicated. 

• When the triangle is full, the barrier is removed and both triangles are 

compared. 

• Encourage discussion about any differences. 


Vary the game by requiring the pattern block pieces to be aligned in the same 

manner. 

Records of the various designs will be displayed so students can compare a 

wide range of solutions. 




1. Using a variety of pattern 

blocks each time, fill the 

equilateral triangle. Make as 

many variations as possible. 

2. Discover whether it is 

possible to use the square or 

the tan rhombus. Explain. 

3. Use the most or fewest 

number of pattern blocks to 

fill the triangle. 

4. Fill the triangle by using 

pattern blocks of two colours 

only. 

5. Use three colours only. 

6. Try to fill the triangle using all 

of the pattern blocks. 

7. Estimate how many green 

triangles are needed to 

exactly fill the triangle. 

8. Start with the yellow 

hexagon and use pattern 

blocks of three colours. 

Is it possible to use two 

hexagons? Why? 



Display Copy 

9. Fill the triangle using six 

blocks and two colours. 

Invent other interesting 

challenges. 

Two possible solutions: 



58 






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59 


D T E S 

Exploring hexagons 

Purpose 

To encourage students 

to use their spatial skills 

(reflection, rotation) to work 

mathematically to create 

hexagons. 

Watch your language 

Throughout our teaching 

relationships with students, 

a major problem frequently 

occurs: the meaning of words. 

Here is one which needs careful 

consideration: 

regular – 

regular fries 

regular (meaning punctual) 

regular army 

regular (meaning periodic, 

rhythm, seasonal) 

regular (as in health 

measurement) 

I am sure you can add to this 

list. Now consider entries in a 

dictionary: 

regular: usual, conforming, 

orderly, even, steady, happening 

at fixed times, following a rule or 

procedure … 

There are fourteen meanings 

of regular in my ‘essential’ 

dictionary. 

So how does a student feel 

when we refer to this shape 

as a regular hexagon? And 

then another regular shape 

as an equilateral triangle and 

as a square, rather than a 

regular quadrilateral? 

Simply, learning the vocabulary 

is possibly as difficult as coming 

to grips with the mathematics in 

and of those shapes. 

1 

3 

This is a regular hexagon. All the sides are 

the same length and all the angles are the 

same size. 

120º 

120º 120º 

120º 120º 

Not all polygons are regular 

120º 

2 

These shapes are hexagons. Each shape 

has six sides. 

These are the same shapes. One hexagon may be reflected (flipped over) to fit exactly on 

the other one. 



Display Copy 

A hexagon has six sides 

and the sum of all the 

angles is 720º. 

90º 90º 

210º 

210º 

60º 60º 



60 


4 

5 

6 

By combining two pattern blocks each time, create a collection of hexagons. 

Many students are familiar with convex polygons. Many of these hexagons are concave. 

This hexagon is 

concave. Note how you 

can draw a line from 

one side of the hexagon 

to another. You cannot 

do this with a convex 

polygon such as the 

yellow pattern block 

hexagon. 



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The ideas behind beehive design in the side bar may be explored using paper strips. 

Starting with three paper strips (at least 24 cm long and 5 cm wide), fold to form a triangle, 

square and hexagon. Compare the area enclosed by each. Note the perimeters will be the 

same on each shape, but the areas will differ. The hexagon will enclose the largest area. 

Did you know? 

The hexagon is an important 

shape in nature. Beehives 

are constructed from regular 

hexagons. Why hexagons? 

There are several reasons. One 

is that hexagons may be easily 

tessellated, but then again so 

may equilateral triangles and 

squares—as you no doubt have 

discovered. 

A hexagon with the same 

perimeter as an equilateral 

triangle or square will enclose a 

much larger area, meaning that 

more honey may be stored in it. 

It also means the bees use less 

material and waste less energy 

building a hexagonal structure. 

Engineers have also found 

a structure built from 

hexagons is very strong. Many 

mathematicians have examined 

beehive structure and found it to 

be very close to the best possible 

structure that could be built for 

the purpose. Amazing isn’t it? 



61 


D T E S 

Perimeter 

Purpose 

To determine the perimeter 

of a figure made with pattern 


1 

The distance around is easy with pattern blocks 

Create this figure on the overhead projector or instruct students to make the model. 

Perimeter with pattern 

blocks 

‘Peri-meter’ literally means ‘to 

measure around’. 

Perimeter: the length of the 

boundary around a closed 

shape. This shape may be 

curved. The perimeter of a circle 

is called a circumference. 

Area: the amount or size of a 

surface. 

Initially, we found it best to 

focus solely on developing the 

concept of perimeter in depth. 

The clever design of the pattern 

blocks provides an easy tool 

for measuring perimeter. With 

the exception of the trapezium, 

all sides of the pattern blocks 

shapes measure the same 

unit. The base of the trapezium 

measures two units. Thus it is 

easy to determine the perimeter 

of any pattern block shape. 

As students experiment with 

the same group of blocks they 

will realise that the length of 

the perimeter may change but 

the area (amount of surface) 

will remain the same, clearly 

demonstrating there is no 

relationship between the 

concepts ‘perimeter’ and ‘area’. 

2 

Determine the length of the perimeter. (18 units) 

Ensure that students know the meaning of ‘perimeter’. 

Rearrange the same pattern blocks to make: 

• a longer perimeter 

• a shorter perimeter 

True or false? 

The more edges that are joined when using 

the same number of the same blocks, the 

shorter the perimeter will be. 

Use these blocks to test the theory. 

Note: The length of 

a side of a pattern 

block is one unit. 

The trapezium has 

one side of two 

units. 





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Students create as many different 

arrangements as possible then 

arrange them in an agreed order. 

Students draw their arrangements 

in the picture frames, ready for 

display. 

Make a display as a discussion 

starter to reinforce the 

fundamental idea that the more 

edges that are joined the shorter 

the perimeter. 

3 

Use eight pattern blocks to show the 

shortest and longest perimeter. 



62 


Perimeter: 

Number of joins 

units of length 



Perimeter: 





Display Copy 


Perimeter: 


Perimeter: 




Enlarge to A3. 



63 


D T E S 

Perimeter and area 

Purpose 

To develop the understanding 

that perimeter and area are 

not related. 

1 

Making sure of the difference 

one square unit of area 

A cautionary warning 

You will have observed that 

no mention has been made of 

the formulas for calculating 

perimeter and area. This 

is deliberate because our 

main aim is to create an 

understanding of the differences 

between the two concepts. Our 

experience has shown that 

confusion is not created when 

approached in the manner 

outlined here. 

As students gain experience, 

it is fair to ask them to design 

rules which will work. Hopefully, 

they will discover that 2 (l + 

w) will work in certain specific 

situations and that l x w will 

not work for a circle. Sounds 

obvious, but countless students 

leave mathematics classes 

in utter confusion about the 

concepts of area and perimeter. 

This occurs because they are 

rushed into the symbolic stage; 

that is, the use of formulas. 

CHALLENGE 

ANSWERS 

1. 2. 

3. 4. 

5. 6. 

2 

3 

Perimeter = 14 units of length 

Area = 12 square units 

Consider the oblong illustrated above. (An oblong is a rectangle whose adjacent sides are 

unequal; so therefore it is a quadrilateral. A rectangle may be a square or an oblong.) 

Create other shapes. 



one unit of length 



Note how the area remains constant, while the perimeter changes. Challenge the students 

to make all the shapes they can with an area of 24 square units. 



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CHALLENGES 

Arrange a 3 x 3 array of pattern blocks orange squares. 

Perimeter = 12 units of length, area = 9 square units. 

1. Reduce the area by one square unit but maintain the 

perimeter at 12 units. 

2. Reduce the area to 7 square units while maintaining the perimeter at 12 units. 

3. Reduce the area to 6 square units while keeping the same perimeter. 

4. Reduce the area to 5 square units while keeping the perimeter the same. 

5. Reduce the area to 4 square units while maintaining the perimeter at twelve units. 

6. Reduce the area to three square units while maintaining the perimeter at 12 units. 

Once the students understand this interesting challenge, create similar challenging 

situations. 

Note: Answers will vary. 



64 


Perimeter patterns 

D T E S 

1 

Discovering by using the same shape to make new shapes 

Tabulate the results. 

Purpose 

To investigate number 

patterns derived from 

situations. 

2 

1 triangle 

3 units of length 

3 triangles 


Investigate this series 

2 triangles 


4 triangles 


3 Arrange efficient seating around 

trapezium-shaped tables. (In some 

places, they are known as trapezoid 

tables.) 

One table may sit 5 students. 

4 

Number of Perimeter 

triangles 

1 3 

2 4 

3 5 

4 6 

Predict the perimeter for 20 triangles. 

Explain how you arrived at your answer. 

Study the perimeter patterns when using 

only the hexagon shape. 



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Pattern predictions 

Predicting with confidence—not 

always with total accuracy—is 

a vital mathematical experience 

for students at all levels. An 

even stronger feature of the 

mathematical experience is to 

carefully consider that prediction 

to see if it will stand or not. We 

believe that testing a prediction 

(hypothesis) is a most important 

mathematical function. 

So, do not hesitate to challenge 

students with oral problems such 

as: 

• Arrange four blue rhombuses 

with an area of 4 units and a 

perimeter of 10 units. 

Expect students to explain 

their answers. 

• I have 12 orange squares. 

Arrange them so that the 

shortest possible fence may 

be built to surround all the 

squares. 

Line up two tables to sit 8 students. 

Continue this pattern to sit 20 students. 

Is there a more efficient way to arrange 

the tables? Explain. 

Study the figures very closely to 

determine how many students could be 

seated at 50 trapezium tables. 

Number of Perimeter 

hexagons 

1 6 

2 10 

3 14 

Make a rule to generalise 

this pattern. Will the same 

rule apply if the hexagons 

are aligned like this? 

Why? 



65 


Trading 

games 

R.I.C. 

Publications 

triangl 

le 

® 

h exagon trapeziu 

m rhombus 

attern block tradi ing 

boar 

rd 

P n 



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When these games are played 

with a sense of fun and 

adventure, students do not 

realise the depth of 

mathematical understanding 

in which they are involved.

Working with other bases assists in 

the understanding of number and 

operations using number. Richard 

Skemp defended the use of bases 

other than 10. He also recognised the 

significance of subitising as students 

developed their numerical understanding. 

Base 10 involves the same concepts as 

those used in bases 2, 4, 5 ... but 10 is too 

big to subitise. So manipulations which 

can be done perceptually for bases up to 5 

depend on counting when working with base 

10 (p. 41). 

Skemp, R. (1989). Structured activities for primary mathematics; how to 

enjoy real mathematics. London. Routledge. 

Creating an environment in which students will feel confident in ‘doing’ 

mathematics is one of our major aims in teaching primary school-age 

students. For it is through the situations we have created that students 

can develop and display their true thinking abilities. 

When students are given the opportunity to use their mathematical 

thinking to construct their ideas, they display an autonomy rarely seen 

in more conventional classroom situations. 

Not only are the students utilising past experiences, but also their 

patterns of problem-solving skills flow into other areas of the 

curriculum. Trading games are full of ‘overflow’ potential. 

These organisational tips will assist you 

For the first few introductory sessions with a group of students, 

organise whole-class exercises. Younger students may sit in a circle 

on the floor so you can observe how they are handling matters. 

Usually, older students do not need such close supervision, but we 

prefer to know the majority of the class has attained in-principle level 

of understanding of the games. 

Initially, teach the students using the simple ‘trading board’. 

Copy and laminate sufficient boards for every student. One teacher 

introduced the games on photocopied sheets. When the students 

knew the games, they were invited to decorate a ‘new’ trading board 

before it was laminated. These boards became personal property 

which the students could take home to teach their parents how to 

trade. 

Once familiar with the games, the students may break into groups 

of three or four. For a series of games, one of the students may be 

appointed ‘the banker’. All transactions must be passed through 

the banker. Personal ‘bank books’ may be kept, but do not insist on 

formal presentation of the transactions. In time, every student will have 

a turn at being ‘the banker’. 

Negotiate with the teacher of another class to have some of your 

‘banker-type’ students teach the games to that class. ‘Bankers’ may 

wear a badge which entitles them to ‘bank’ (teach) for a week. 

Once the games are well-known, play for 10 minutes before a break. 

Keep a progressive score. Largest total on Friday wins some sort of 

honour! 

Introduction to trading games 

Long before you introduce formal materials, like Base 

10 blocks and the spiked abacus, you need to foster 

and develop the concepts of ‘place value’, the ‘four 

operations’ and number facts’ by playing these 

trading games. As one excited teacher said: ‘The 

students find these games so easy and enthralling. 

I’m sorry I haven’t used them before’. 

When these games are played with a sense of fun 

and adventure, the students do not realise the 

depth of mathematical understanding in which 

they are involved. And, as the students begin 

to formalise those understandings, you will 

be amazed at the number understanding the 

students have developed. 

We recommend your first adventure into 

trading games begins after the students 

have been at school for about six to eight 

months, depending on how they are 

travelling the ‘school-road’. Once the 

students have mastered the concept 

of trading, leave the idea for about 

three weeks. During this time, the 

ideas will blend into the student’s 

experiences and when you return 

with a ‘Remember when …’ the 

students will quickly recall their 

trading activities. 



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67

D T E S All yellow 

Purpose 

To develop trading based 

on the area relationship 

between pattern blocks. 

Trading games 

Trading games need not be 

confined solely to Unifix ® or 

Base 10 blocks. Any material 

which shows a relationship may 

be used to form the base of a 

trading game. Pattern blocks 

make an ideal medium for 

playing trading games. 

The numerical base is 6: 

6 triangles are equivalent to the 

hexagon (a trade of 6 for 1) 

3 blue rhombuses are 

equivalent to the hexagon (a 

trade of 3 for 1) 

2 trapeziums may be traded for 

one hexagon (a trade of 2 for 1) 

Encourage the students 

to discover all the trading 

relationships. 

Most of the ideas presented 

in this book are based on a 

relationship concept. In fact, 

we suggest that the notion of 

relationships is one of the key 

factors in the process of the 

mathematical experience. When 

considering the numbers 1 

and 2, think of the relationship 

factors involved; for example, 

2 is larger than 1, in fact twice 

as large. 

We encourage students to talk 

(E) each move. When ‘earning’ 

triangles we call the process 

composing (addition) so when 

the ‘tax’ moves are played we 

call that process decomposing 

(subtraction). Young students 

show no difficulty in using 

composing and decomposing 

competently so our advice is 

to use sound and consistent 

vocabulary. 

You will need 

1 hexagon board each 

blue rhombus and yellow 

hexagon pattern blocks 

Method 

• Two to four players take turns to roll the die and collect blue pattern blocks for the 

number rolled. 

• The player places the blocks on one or more hexagons on the board. When a hexagon is 

fully covered, the player may trade three blue rhombuses for one yellow hexagon. 

• The first player to fill his/her board with yellow hexagons is the winner. 

Play this game using other pattern block shapes; for example, trade six green triangles for 

one hexagon. 

Variation 1 

Each hexagon must be filled with three 

colours, except when a six is thrown. 

triangle................represents..........1 

blue rhombus.....represents..........2 

trapezium...........represents..........3 

hexagon.............represents..........6 

• On each throw of the die, students 

collect the appropriate shapes. For 

example, if a four is thrown a player 

may choose two blue rhombuses 

or a trapezium and a triangle or four 

triangles. 


1–6 die 



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Variation 2 

Play the game backwards. 

• Fill the board with yellow hexagons. 

• When the triangle has a value of 1, 

remove the value thrown each time. 

This will involve trading skills. 

• The first player to clear the board 

is the winner. (To keep the game 

alive, we suggest the number of 

hexagons may be limited to 7.) 

• When a six is thrown a player 

collects a hexagon. 



68 





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69 


D T E S 

Purpose 

To develop concepts of 

addition and subtraction. 

Trade with pattern 


Create thoughtful arithmetical challenges 

Earning pattern block money 

The trading rules: 

Use these pieces only. 





Throw a die to collect 

triangles and place them 

in the appropriate column 

on the trading board. 

However, in the triangle 

column only one triangle 

is allowed because two 

triangles can be traded 

for a blue rhombus and 

a triangle and a blue 

rhombus are equivalent to 

a trapezium. 

The tax game 

On every third throw 

use a different coloured 

die. This represents ‘tax’ 

and hence the throw 

must be paid back 

to the bank in green 

triangles. If a player 

does not have enough 

triangles, a trade ‘down’ 

(decompose) will be 

necessary. 

The bonus game 

Use a 1–4 (tetrahedron) 

die. The throw will 

indicate the bonus: 

increase the balance of 

pattern blocks on the 

trading board by 1, 2, 3 

or 4 times according to 

the throw. Record the 

trading process. 

The opposite may be 

played: the loss game. 

Divide the collection 

into 2, 3, or 4 equal 

parts. Any remainder 

may be retained by the 

player. 



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Students will soon learn that five triangles may be traded for a trapezium and a rhombus. 

Aim to have the minimum number of blocks at any one time. 

Play to achieve an agreed target. 



70 


Pattern block trading board 

hexagon trapezium rhombus triangle 




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71 


D T E S 

Purpose 

Strengthening arithmetic 

skills. 

Shopping with pattern 


Creative number operation skills 

A South Pacific island nation has decided to use shells as their form of currency. 

Strengthening arithmetic 

skills 

We have discovered that 

arithmetic skills flow so much 

more easily when supported 

by appropriate manipulative 

material. The visual and/ 

or concrete representations 

strengthen a student’s intuitive 

skills, a process often ignored 

as mathematical ideas are 

developed. I frequently play 

‘Guess what’s in my pocket’. 

A green triangle has a value of 

6. ‘In my pocket I have blocks 

which show half of 72 less six. 

What blocks might I have?’ 

Most times the challenge is met 

eagerly. 

A few decades ago, these ideas 

were introduced under the broad 

heading of value-relations. One 

of the natural results of this 

thinking is the appreciation of 

fraction, ratio and percentage 

concepts. By rushing into this 

conceptual area without the 

preceding experiences, as 

described on pages 25–26 and 

66–67, both students and many 

teachers become confused. 

Consider the mathematical 

procedures being reinforced 

and the number skills being 

developed in these pattern 

block shopping games. Our 

experience has shown that 

students become really involved 

in these make-believe shopping 

experiences. 

1 

2 

4 

Different ways to show 20 shells 

with two blocks with three or more blocks 

10s + 10s 15s + 5s 5s + 5s + 5s + 5s 10s + 5s + 5s 

The 100 shell collection 

Four blocks are used 

30s + 30s + 30s + 10s 

Five blocks are used 

30s + 30s + 15s + 15s + 10s 

Make 100 shell models using 4, 5, 6 and 12 

blocks in a model. 

3 Change the values of the pattern blocks and 

carry out similar activities to the above: 

Example: 

Challenges 

• You have 10 hexagons (10H) and you 

purchase a 3 kg bag of sugar and 2 kg of 

apples. What change will you receive? 

• Detergent is half price today. How much 

will a bottle cost? 

• Purchase three items so that you have 

almost no change remaining from 10 

hexagons (10H). 

• Plan a business venture marketing toffee 

apples. (I kg of apples needs about 

2 kg of sugar to make a suitable toffee.) 

No student may invest more than 2H 

to finance the business. How many 

shareholders will be needed? 

If 

= 5 shells (s) 



Cut along all dotted lines, fold all 

solid lines so that each grocery 

item stands up. 

There are two distinct activities on 

these sheets. 

1. Grocery items with illustrated 

pattern block prices 

2. The same grocery items with a 

symbolic code to indicate the 

prices. The code will be known 

because it is introduced in 

pattern block trading. 

• Decide on the value of the 

triangle before visiting the 

shop. 



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6s 3s 2s 1s 

• Grocery items may be cut out 

to make stand-up figures. This 

will help the students sort the 

various items. 

• Encourage the collection of 

used grocery containers to 

make a class shop. Students 

may translate prices from 

Shells to pattern block money; 

for example, an item that 

costs 210s will cost 3H1Tr (3 

hexagons and a trapezium) in 

pattern block money when a 

triangle = 10s. 

• Challenge students to create 

a pattern block shop ‘special’ 

flyer. 

• Invent a simple code to 

describe the prices. 



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73 


Shared 

sometimes 

ideas 

Best success in any learning situation 

is achieved when all participants feel 

they are part of the procedure. The 

great teacher manages to get people 

involved and this requires both skill and 

patience. Patience, because the learning 

must come from within the learner and 

this takes a while to occur. 

Switch roles—be a learner! 

One of the techniques we use to engage students is the ‘surprise’ reaction. 

Although the idea may be well-known to you, the teacher, when a student 

displays the ‘look what I have discovered’ behaviour, reacts by saying 

something like ‘Wow! What a good thought. I need to know about 

that, how did you discover that?’ Eagerly, the student will take on an 

explaining role. You will show interest then challenge the student to 

demonstrate the idea to another group. Never mind if all the students 

in the class do not hear the explanation. ‘Word’ will pass around! 

Deliberate mistakes 

A most effective technique to foster intense discussion is to 

‘secretly’—unobserved by the students—create an error. Do 

not make it too obvious. It could be with the blocks or verbally. 

The students think the creation is good but you, the teacher, 

disagree, and the students will be challenged to explain. Finding 

the ‘wrongness’ is an indelible learning experience. 

Students become instructors 

When a student ‘knows’ invest that student with the privilege 

of teaching/showing that idea to others. It does not have to 

be in the same class. In one school, students were licensed to 

teach in other Years, providing it was okay with the teacher. 

Create a pamphlet or small magazine 

With easy access to publishing programs and the ability of so 

many young students to ‘create’ on the computer, encourage 

students to develop information packages to share with other 

people, especially parents. 



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Many students have the ability to create PowerPoint 

presentations. Use these skills to share ideas in a classroom. 

Pattern block art gallery 

Excellent creations may be photographed, then enlarged to fit a large 

picture frame. These attractive ‘pictures’ may be displayed as an art 

exhibition. In fact, enlist the assistance of the art teacher—no better way 

to share! 

Long-term displays 

Create a space in the classroom where models may be displayed for some 

time. Students will have the opportunity to share ideas with others. Use 

firm cardboard or plywood as a base on which to build models for ease of 

movement. 



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Photocopiable resources 

Pages 75–78 provide a collection of 

pattern block grids for students to 

colour, cut and arrange. 

We suggest that about 20 copies of 

each pattern block grid are printed so 

students are able to make a free choice 

as to which grid to use. Alternatively, a 

large quantity of the all-purpose pattern 

block grid (p. 76) is made available. 

All shapes (except the square) may 

be reproduced on this grid easily. 

More experienced students may be 

challenged to design their own grids. 

Pattern block portfolio 

Pattern blocks will be beneficial in all primary 

school classes. It is a good idea for the 

student to have the opportunity to see his/ 

her growth in experience over the years. We 

suggest that a pattern block portfolio be 

started during a student’s first experience 

with the material. The portfolio ‘travels’ with 

the student through the school. The portfolio 

may include: 

• designs created and designs converted 

to pattern 

• finding angles and rotations 

• symmetry 

• photographs of mirror explorations 

• opinion and assessment page. 

Colour, cut and paste 

Students create a set of paper pattern blocks. By 

creating interesting patterns, tessellation skills will 

be reinforced. 

Pattern Block Man 

Photocopy the all-purpose grid sheet (p. 76), on 

red or coloured paper. With the pattern blocks, 

create Pattern Block Man. Copy the model by 

gluing paper rhombus pieces on a background 

which has been drawn or painted to provide 

an appropriate setting. (A good art session.) A 

student keeps the most satisfying creation for 

inclusion in the pattern block portfolio. 

Students may also create a poster to show 

symmetry with pattern blocks. 

Specific block 

patterns 

Students use two or 

three pattern blocks to 

create a design on one 

of the grids. A display 

of the finished products 

will provide a few 

surprises! 



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Answers 

How old is Pattern Block Man? (p. 9) 

Card 1 

Card 2 

Card 3 

30 years old (30 triangles) 

60 years old (10 trapeziums) 

daughter (16) son (12) 

man (40) woman (34) 

Card 4 

Teacher check 

Giving the shapes value (pp 29–31) 

Card 1 

1. Answers will vary; 3 hexagons + 3 trapeziums 

2. 12 

3. Answers will vary; 2 trapeziums and 2 hexagons 

4. 6 hexagons 

Card 2 

1. triangle = 2, blue rhombus = 4, trapezium = 6 

2. trapezium 

3. Answers will vary; 

6 = 1 trapezium, 12 = hexagon, 

18 = 1 trapezium + 1 hexagon, 

24 = 2 hexagons 

30 = 2 hexagons + 1 trapezium 

36 = 3 hexagons 

4. 5 

Card 3 

1. 1 hexagon + 1 trapezium + 1 blue rhombus 

2. Teacher check 

3. Answers will vary; 2 trapeziums + 1 triangle 

4. 13 (5 x 2 + 3) 

Card 4 

1. triangle = 1 / 2 

, blue rhombus = 1, 

trapezium = 1 1 / 2 

2. 5 = hexagon + trapezium + triangle 

3. 5 1 / 2 

, 3 blocks: 1 hexagon, 1 trapezium and 1 

blue rhombus 

4. 7 

Card 5 

1. Answers will vary 

2. 6 

3. Answers will vary; 5 hexagons + 2 trapeziums 

4. 6 hexagons 

Card 6 

1. Answers will vary 

2. 3 x 2 = 3 blue rhombuses 

5 x 6 = 5 hexagons 

7 x 1 = 7 triangles 

2 x 3 = 2 trapeziums 

3. A fence panel = 

Card 7 

1. (a) triangle = 1 / 6 

, blue rhombus = 2 / 6 

or 1 / 3 

, 

trapezium = 3 / 6 

or 1 / 2 

(b) 1 

(c) 2 

2. 3 x 1 + 2 x 1 / 2 

+ 7 x 1 / 6 

3. hexagon + trapezium 

5 

/ 6 

(5 triangles) + 4 / 6 

(2 blue rhombuses) = 9 / 6 

= 

1 1 / 2 

= 1 (1 hexagon) + 1 / 2 

(1 trapezium) 

Card 8 

1. (a) triangle = 1 / 2 

, blue rhombus = 1, 


(b) 6 1 / 2 

2. 16 1 / 2 

= 3 x 3 + 3 x 1 1 / 2 

+ 3 x 1 

Card 9 


, hexagon = 3, 


2. triangle 

3. Answers will vary; 3 = hexagon, 6 = 2 hexagons 

etc. Reasons will vary 

4. 26 

Card 10 


, blue rhombus = 1 / 3 

; 

Note: 1 / 6 

+ 1 / 3 

= 1 / 2 

total value = 3 5 / 6 

2. 3 hexagons 

3. 8 trapeziums + 4 triangles + 1 blue rhombus 

Card 11 

1. Answers will vary; 1 2 / 3 

= 5 triangles, 

3 1 / 3 

= 1 hexagon + 1 trapezium + 1 triangle 

2. 1 hexagon + 1 trapezium. Explanations will vary. 

3. Answers will vary; 2 hexagons + 2 blue 

rhombuses 

4. 10. Explanations will vary 

5. Answers will vary; 2 2 / 3 

= hexagon + blue rhombus 

3 1 / 3 

= hexagon + trapezium + triangle 

Card 12 

1. trapezium 

2. 3 hexagons 

3. True; a ratio of 1:8 

4. 1 hexagon 

Pattern block fractions (p. 33) 

Answers will vary; 

Card 1 1 = 2 hexagons + 1 trapezium 

Card 2 1 = 2 hexagons 

Card 3 1 1 / 2 

= 3 hexagons 

Card 4 1 2 / 3 

= 3 hexagons + 1 blue rhombus 

Card 5 2 

/ 9 

= 2 triangles or 1 blue rhombus 

Card 6 1 1 / 2 

= 2 hexagons, 1 trapezium + 1 triangle 

Exploring angles (p. 44) 

1. (a) 60º (b) 60º (c) 30º (d) 30º 

(e) 90º (f) 150º 

2. (a) 120º obtuse (b) 60º acute 

(c) 120º obtuse (d) 90º right 

(e) 150º obtuse (f) 150º obtuse 

(g) 210º reflex 



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80 


Acute angle – An angle between 0º and 90º. 

Angle – Technically, an angle is the union of two rays with a common end point. 

The size of the angle depends on the amount of rotation from one ray to 

the other. 

Axis/Line of 

symmetry – The line that divides a shape in half so it can be reflected onto itself. 

Sometimes referred to as a mirror line as one half is the mirror image of 

the other. 

Congruent – Two shapes are congruent if they are the same size and shape. 

Equilateral 

triangle – A triangle in which all the sides are the same length and the angles are 

the same size. 

Hexagon – A polygon with six sides. 

Obtuse angle – An angle between 90º and 180º. 

Order of rotational 

symmetry – Refers to the number of times a shape may be rotated on itself. 

Quadrilateral – A four-sided polygon. 

Reflection – Flipped over a line—mirror image. 

Reflex angle – An angle between 180º and 360º. 

Regular polygon – 

A polygon with all sides and angles congruent. 

Rhombus – A quadrilateral with four congruent sides. 

Rotation – Turned about a point. 

Rotational 

symmetry – The number of ways a shape can be rotated to fit on itself. That is, a 

shape can be turned part way around and look the same. This matches 

the number of lines of symmetry. 

Similar – Two figures are similar if they are exactly the same shape. 



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Tessellation – A pattern of congruent shapes that cover (tile) a surface without leaving 

any gaps. 

Glossary 

Translation – Moving a figure in such a way that all the points move the same distance 

in the same direction. 

Trapezium – A quadrilateral with a pair of parallel sides. 

We would like to thank Linda Marshall for her assistance. 

Our sincere thanks to the staff, children and parents at Dalyellup 

Beach Primary School. And special thanks to Adrian, Daniel, 

Celine, James, Sabrina and Leighland for their assistance. 

Over the years, we have been inspired to use materials by 

educationalists such as Mary Baratta Lorton, Pamela Leibeck and 

Kathy Richardson. 



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RIC-6001 Developing mathematics with pattern blocks

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