Primary Years Programme

Mathematics scope and sequencePrimary Years Programme

Mathematics scope and sequencePrimary Years Programme
Mathematics scope and sequence

Published February 2009
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PYP110IB mission statement
The International Baccalaureate aims to develop inquiring, knowledgeable and caring young people who help tocreate a better and more peaceful world through intercultural understanding and respect.
To this end the organization works with schools, governments and international organizations to develop
challenging programmes of international education and rigorous assessment.
These programmes encourage students across the world to become active, compassionate and lifelong learners
who understand that other people, with their differences, can also be right.

IB learner profile
The aim of all IB programmes is to develop internationally minded people who, recognizing their common
humanity and shared guardianship of the planet, help to create a better and more peaceful world.
IB learners strive to be:
Inquirers

They develop their natural curiosity. They acquire the skills necessary to conduct inquiry
and research and show independence in learning. They actively enjoy learning and thislove of learning will be sustained throughout their lives.

Knowledgeable

They explore concepts, ideas and issues that have local and global significance. In sodoing, they acquire in-depth knowledge and develop understanding across a broad andbalanced range of disciplines.

Thinkers

They exercise initiative in applying thinking skills critically and creatively to recognize
and approach complex problems, and make reasoned, ethical decisions.

Communicators

They understand and express ideas and information confidently and creatively in morethan one language and in a variety of modes of communication. They work effectively
and willingly in collaboration with others.

Principled

They act with integrity and honesty, with a strong sense of fairness, justice and respect
for the dignity of the individual, groups and communities. They take responsibility for
their own actions and the consequences that accompany them.

Open-minded

They understand and appreciate their own cultures and personal histories, and are open
to the perspectives, values and traditions of other individuals and communities. They areaccustomed to seeking and evaluating a range of points of view, and are willing to growfrom the experience.

CaringThey show empathy, compassion and respect towards the needs and feelings of others.
They have a personal commitment to service, and act to make a positive difference to thelives of others and to the environment.

Risk-takers

They approach unfamiliar situations and uncertainty with courage and forethought,
and have the independence of spirit to explore new roles, ideas and strategies. They arebrave and articulate in defending their beliefs.

Balanced

They understand the importance of intellectual, physical and emotional balance toachieve personal well-being for themselves and others.

Reflective

They give thoughtful consideration to their own learning and experience. They are able
to assess and understand their strengths and limitations in order to support their learningand personal development.

© International Baccalaureate Organization 2007

Contents

Introduction to the PYP mathematics scope and sequence

1

What the PYP believes about learning mathematics

1

Mathematics in a transdisciplinary programme2

The structure of the PYP mathematics scope and sequence3

How to use the PYP mathematics scope and sequence

4

Viewing a unit of inquiry through the lens of mathematics

5

Learning continuums

6

Data handling			6

Measurement			

10

Shape and space			14

Pattern and function		

18

Number				21

Samples				26

Mathematics scope and sequence

Introduction to the PYP mathematics scopeand sequence

The information in this scope and sequence document should be read in conjunction with the mathematics
subject annex in Making the PYP happen: A curriculum framework for international primary education (2007).

What the PYP believes about learning mathematics
The power of mathematics for describing and analysing the world around us is such that it has become ahighly effective tool for solving problems. It is also recognized that students can appreciate the intrinsic
fascination of mathematics and explore the world through its unique perceptions. In the same way thatstudents describe themselves as “authors” or “artists”, a school’s programme should also provide students
with the opportunity to see themselves as “mathematicians”, where they enjoy and are enthusiastic when
exploring and learning about mathematics.
In the IB Primary Years Programme (PYP), mathematics is also viewed as a vehicle to support inquiry,
providing a global language through which we make sense of the world around us. It is intended thatstudents become competent users of the language of mathematics, and can begin to use it as a way of
thinking, as opposed to seeing it as a series of facts and equations to be memorized.

How children learn mathematics
It is important that learners acquire mathematical understanding by constructing their own meaning
through ever-increasing levels of abstraction, starting with exploring their own personal experiences,
understandings and knowledge. Additionally, it is fundamental to the philosophy of the PYP that, since it is
to be used in real-life situations, mathematics needs to be taught in relevant, realistic contexts, rather thanby attempting to impart a fixed body of knowledge directly to students. How children learn mathematics
can be described using the following stages (see figure 1).

Constructing
meaning

Transferring
meaning

Applying with
understanding
Figure 1
How children learn mathematics

Mathematics scope and sequence1

Introduction to the PYP mathematics scope and sequence

Constructing meaning about mathematics
Learners construct meaning based on their previous experiences and understanding, and by reflecting
upon their interactions with objects and ideas. Therefore, involving learners in an active learning process,where they are provided with possibilities to interact with manipulatives and to engage in conversations
with others, is paramount to this stage of learning mathematics.
When making sense of new ideas all learners either interpret these ideas to conform to their present
understanding or they generate a new understanding that accounts for what they perceive to be occurring.
This construct will continue to evolve as learners experience new situations and ideas, have an opportunity
to reflect on their understandings and make connections about their learning.

Transferring meaning into symbols
Only when learners have constructed their ideas about a mathematical concept should they attempt
to transfer this understanding into symbols. Symbolic notation can take the form of pictures, diagrams,
modelling with concrete objects and mathematical notation. Learners should be given the opportunity todescribe their understanding using their own method of symbolic notation, then learning to transfer theminto conventional mathematical notation.

Applying with understanding
Applying with understanding can be viewed as the learners demonstrating and acting on theirunderstanding. Through authentic activities, learners should independently select and use appropriate
symbolic notation to process and record their thinking. These authentic activities should include a rangeof practical hands-on problem-solving activities and realistic situations that provide the opportunity todemonstrate mathematical thinking through presented or recorded formats. In this way, learners are able toapply their understanding of mathematical concepts as well as utilize mathematical skills and knowledge.
As they work through these stages of learning, students and teachers use certain processes of mathematicalreasoning.
•

They use patterns and relationships to analyse the problem situations upon which they are working.

•

They make and evaluate their own and each other’s ideas.

•

They use models, facts, properties and relationships to explain their thinking.

•

They justify their answers and the processes by which they arrive at solutions.

In this way, students validate the meaning they construct from their experiences with mathematicalsituations. By explaining their ideas, theories and results, both orally and in writing, they invite constructive
feedback and also lay out alternative models of thinking for the class. Consequently, all benefit from thisinteractive process.

Mathematics in a transdisciplinary programme
Wherever possible, mathematics should be taught through the relevant, realistic context of the units ofinquiry. The direct teaching of mathematics in a unit of inquiry may not always be feasible but, whereappropriate, prior learning or follow-up activities may be useful to help students make connections between
the different aspects of the curriculum. Students also need opportunities to identify and reflect on “bigideas” within and between the different strands of mathematics, the programme of inquiry and othersubject areas.
Links to the transdisciplinary themes should be explicitly made, whether or not the mathematics is beingtaught within the programme of inquiry. A developing understanding of these links will contribute to thestudents’ understanding of mathematics in the world and to their understanding of the transdisciplinary

2

Mathematics scope and sequenceIntroduction to the PYP mathematics scope and sequence

theme. The role of inquiry in mathematics is important, regardless of whether it is being taught inside oroutside the programme of inquiry. However, it should also be recognized that there are occasions when it ispreferable for students to be given a series of strategies for learning mathematical skills in order to progressin their mathematical understanding rather than struggling to proceed.

The structure of the PYP mathematics scope andsequence
This scope and sequence aims to provide information for the whole school community of the learning that isgoing on in the subject area of mathematics. It has been designed in recognition that learning mathematics
is a developmental process and that the phases a learner passes through are not always linear or age related.
For this reason the content is presented in continuums for each of the five strands of mathematics—data
handling, measurement, shape and space, pattern and function, and number. For each of the strands there
is a strand description and a set of overall expectations. The overall expectations provide a summary of
the understandings and subsequent learning being developed for each phase within a strand.
The content of each continuum has been organized into four phases of development, with each phasebuilding upon and complementing the previous phase. The continuums make explicit the conceptual
understandings that need to be developed at each phase. Evidence of these understandings is describedin the behaviours or learning outcomes associated with each phase and these learning outcomes relatespecifically to mathematical concepts, knowledge and skills.
The learning outcomes have been written to reflect the stages a learner goes through when developing
conceptual understanding in mathematics—constructing meaning, transferring meaning into symbols
and applying with understanding (see figure 1). To begin with, the learning outcomes identified in
the constructing meaning stage strongly emphasize the need for students to develop understanding ofmathematical concepts in order to provide them with a secure base for further learning. In the planning
process, teachers will need to discuss the ways in which students may demonstrate this understanding. The
amount of time and experiences dedicated to this stage of learning will vary from student to student.
The learning outcomes in the transferring meaning into symbols stage are more obviously demonstrable
and observable. The expectation for students working in this stage is that they have demonstrated
understanding of the underlying concepts before being asked to transfer this meaning into symbols. It is
acknowledged that, in some strands, symbolic representation will form part of the constructing meaning
stage. For example, it is difficult to imagine how a student could construct meaning about the way in which
information is expressed as organized and structured data without having the opportunity to collect andrepresent this data in graphs. In this type of example, perhaps the difference between the two stages isthat in the transferring meaning into symbols stage the student will be able to demonstrate increased
independence with decreasing amounts of teacher prompting required for them to make connections.
Another difference could be that a student’s own symbolic representation may be extended to include
more conventional methods of symbolic representation.
In the final stage, a number of learning outcomes have been developed to reflect the kind of actions andbehaviours that students might demonstrate when applying with understanding. It is important to notethat other forms of application might be in evidence in classrooms where there are authentic opportunities
for students to make spontaneous connections between the learning that is going on in mathematics andother areas of the curriculum and daily life.
When a continuum for a particular strand is observed as a whole, it is clear how the conceptual
understandings and the associated learning outcomes develop in complexity as they are viewed across thephases. In each of the phases, there is also a vertical progression where most learning outcomes identified
in the constructing meaning stage of the phase are often described as outcomes relating to the transferring

Mathematics scope and sequence3

Introduction to the PYP mathematics scope and sequence

meaning into symbols and applying with understanding stages of the same phase. However, on someoccasions, a mathematical concept is introduced in one phase but students are not expected to apply theconcept until a later phase. This is a deliberate decision aimed at providing students with adequate time andopportunities for the ongoing development of understanding of particular concepts.
Each of the continuums contains a notes section which provides extra information to clarify certain learning
outcomes and to support planning, teaching and learning of particular concepts.

How to use the PYP mathematics scope andsequence
In the course of reviewing the PYP mathematics scope and sequence, a decision was made to provide schools
with a view of how students construct meaning about mathematics concepts in a more developmental
way rather than in fixed age bands. Teachers will need to be given time to discuss this introduction andaccompanying continuums and how they can be used to inform planning, teaching and assessing ofmathematics in the school. The following points should also be considered in this discussion process.
•

It is acknowledged that there are earlier and later phases that have not been described in thesecontinuums.

•

Each learner is a unique individual with different life experiences and no two learning pathways are
the same.

•

Learners within the same age group will have different proficiency levels and needs; therefore, teachers
should consider a range of phases when planning mathematics learning experiences for a class.

•

Learners are likely to display understanding and skills from more than one of the phases at a time.
Consequently, it is recognized that teachers will interpret this scope and sequence according to theneeds of their students and their particular teaching situations.

•

The continuums are not prescriptive tools that assume a learner must attain all the outcomes of aparticular phase before moving on to the next phase, nor that the learner should be in the same phasefor each strand.

Each teacher needs to identify the extent to which these factors affect the learner. Plotting a mathematicalprofile for each student is a complex process for PYP teachers. Prior knowledge should therefore never beassumed before embarking on the presentation or introduction of a mathematical concept.
Schools may decide to use and adapt the PYP scope and sequences according to their needs. For example,
a school may decide to frame their mathematics scope and sequence document around the conceptual
understandings outlined in the PYP document, but develop other aspects (for example outcomes, indicators,
benchmarks, standards) differently. Alternatively, they may decide to incorporate the continuums from the
PYP documents into their existing school documents. Schools need to be mindful of practice C1.23 in the
IB Programme standards and practices (2005) that states, “If the school adapts, or develops, its own scopeand sequence documents for each PYP subject area, the level of overall expectation regarding student
achievement expressed in these documents at least matches that expressed in the PYP scope and sequence
documents”. To arrive at such a judgment, and given that the overall expectations in the PYP mathematics
scope and sequence are presented as broad generalities, it is recommended that the entire document beread and considered.

4

Mathematics scope and sequenceIntroduction to the PYP mathematics scope and sequence

Viewing a unit of inquiry through the lens ofmathematics
The following diagram shows a sample process for viewing a unit of inquiry through the lens of mathematics.
This has been developed as an example of how teachers can identify the mathematical concepts, skills andknowledge required to successfully engage in the units of inquiry.
Note: It is important that the integrity of a central idea and ensuing inquiry is not jeopardized by a subjectspecific focus too early in the collaborative planning process. Once an inquiry has been planned through to
identification of learning experiences, it would be appropriate to consider the following process.
Will mathematics inform this unit?
Do aspects of the transdisciplinary theme initially stand out as being mathematics related?
Will mathematical knowledge, concepts and skills be needed to understand the central idea?
Will mathematical knowledge, concepts and skills be needed to develop the lines of inquiry within the unit?

What mathematical knowledge, concepts and skills will the students need to be able to engage with and/or
inquire into the following? (Refer to mathematics scope and sequence documents.)•

Central idea•

Lines of inquiry•

Assessment tasks•

Teacher questions, student questions•

Learning experiences

In your planning team, list the knowledge, concepts and skills.

What prior knowledge, concepts and skills do the students have that can be utilized and built upon?
Which stages of understanding are the learners in the class working at—constructing meaning, transferring
meaning into symbols, or applying with understanding?

How will we know what they have learned? Identify opportunities for assessment.

Decide which aspects can be learned:•

within the unit of inquiry (learning through mathematics)•

as subject-specific, prior to being used and applied in the context of the inquiry (inquiry into
mathematics).

Figure 2
Sample processes for viewing a unit of inquiry through the lens of mathematics
Mathematics scope and sequence5

Learning continuums

Data handling
Data handling allows us to make a summary of what we know about the world and to make inferences
about what we do not know.
•

Data can be collected, organized, represented and summarized in a variety of ways to highlightsimilarities, differences and trends; the chosen format should illustrate the information without bias
or distortion.

•

Probability can be expressed qualitatively by using terms such as “unlikely”, “certain” or “impossible”. It
can be expressed quantitatively on a numerical scale.

Overall expectations
Phase 1
Learners will develop an understanding of how the collection and organization of information helps tomake sense of the world. They will sort, describe and label objects by attributes and represent informationin graphs including pictographs and tally marks. The learners will discuss chance in daily events.

Phase 2
Learners will understand how information can be expressed as organized and structured data and that thiscan occur in a range of ways. They will collect and represent data in different types of graphs, interpreting theresulting information for the purpose of answering questions. The learners will develop an understandingthat some events in daily life are more likely to happen than others and they will identify and describe
likelihood using appropriate vocabulary.

Phase 3
Learners will continue to collect, organize, display and analyse data, developing an understanding of
how different graphs highlight different aspects of data more efficiently. They will understand that scale
can represent different quantities in graphs and that mode can be used to summarize a set of data. The
learners will make the connection that probability is based on experimental events and can be expressed
numerically.

Phase 4
Learners will collect, organize and display data for the purposes of valid interpretation and communication.
They will be able to use the mode, median, mean and range to summarize a set of data. They will create andmanipulate an electronic database for their own purposes, including setting up spreadsheets and using
simple formulas to create graphs. Learners will understand that probability can be expressed on a scale (0–1or 0%–100%) and that the probability of an event can be predicted theoretically.

6

Mathematics scope and sequenceMathematics scope and sequence•
identify, read and interpret range andscale on graphs

identify, describe and explain therange, mode, median and mean in aset of datacollect, display and interpret data incircle graphs (pie charts) and line graphs•

collect and represent data in different
types of graphs, for example, tallymarks, bar graphs

sort and label real objects byattributes.

understand that probability is basedon experimental events.

••

understand that one of the purposesof a database is to answer questions
and solve problems••

understand the difference between
experimental and theoreticalprobability.

•

understand that the mode can beused to summarize a set of data•

understand the concept of chance indaily events (impossible, less likely,
maybe, most likely, certain).

•collect, display and interpret datausing simple graphs, for example, bargraphs, line graphs

understand that probability can be
expressed in scale (0–1) or per cent
(0%–100%)•

understand that scale can represent
different quantities in graphs•

discuss chance in daily events(impossible, maybe, certain).

••understand that the mode, median,mean and range can summarize a setof data•understand that information about
themselves and their surroundings
can be collected and recorded indifferent ways•

understand that information about
themselves and their surroundings
can be obtained in different ways••

understand that different types ofgraphs have special purposes•

understand that data can becollected, displayed and interpretedusing simple graphs, for example, bargraphs, line graphs•

understand that sets can be organized
by one or more attributes•

understand that sets can be organizedby different attributes•

represent information throughpictographs and tally marks

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:•

The probability of an event can bepredicted theoretically.

Probability can be expressed in numericalnotations.

When transferring meaning into
symbols learners:

Probability can be represented on a scale
between 0–1 or 0%–100%.

Probability can be based on experimental
events in daily life.

Some events in daily life are more likely tohappen than others.

Events in daily life involve chance.

When transferring meaning into
symbols learners:

Range, mode, median and mean can beused to analyse statistical data.

Different graph forms highlight different
aspects of data more efficiently.

Objects and events can be organized indifferent ways.

Organizing objects and events helps us tosolve problems.

When transferring meaning into
symbols learners:

Conceptual understandings
Data can be presented effectively for validinterpretation and communication.

Conceptual understandings
Data can be collected, organized,
displayed and analysed in different ways.

Conceptual understandings
Information can be expressed asorganized and structured data.

Conceptual understandings
We collect information to make sense of
the world around us.

When transferring meaning into
symbols learners:

Phase 4Phase 3Phase 2Phase 1

Learning continuum for data handling

Learning continuums

78

identify and describe chance in daily
events (impossible, less likely, maybe,
most likely, certain).

•*Living graphs refer to data that isorganized by physically moving andarranging students or actual materials
in such a way as to show and compare
quantities.

Students should have a lot of experienceof organizing data in a variety of ways,
and of talking about the advantages anddisadvantages of each. Interpretations ofdata should include the information thatcannot be concluded as well as that whichcan. It is important to remember thatthe chosen format should illustrate theinformation without bias.

Students should have the opportunity
to use databases, ideally, those createdusing data collected by the students thenentered into a database by the teacher ortogether.

Notes
Using data that has been collected andsaved is a simple way to begin discussing
the mode. A further extension of mode is
to formulate theories about why a certain
choice is the mode.

Notes
A database is a collection of data, where
the data can be displayed in many forms.
The data can be changed at any time. A
spreadsheet is a type of database where
information is set out in a table. Usinga common set of data is a good way forstudents to start to set up their own
databases. A unit of inquiry would be
an excellent source of common data forstudent practice.

use tree, Venn and Carroll diagrams to
explore relationships between data

•

Notes
An increasing number of computer andweb-based applications are available thatenable learners to manipulate data inorder to create graphs.

determine the theoretical probabilityof an event and explain why it might
differ from experimental probability.

•

use probability to determine
mathematically fair and unfair games
and to explain possible outcomes

•

Notes
Units of inquiry will be rich inopportunities for collecting andorganizing information. It may be usefulfor the teacher to provide scaffolds, suchas questions for exploration, and themodelling of graphs and diagrams.

create and manipulate an electronic
database for their own purposes•interpret range and scale on graphs•

describe real objects and events byattributes.

•

express probability using simple
fractions.

identify, describe and explain therange, mode, median and mean in aset of data•

select appropriate graph form(s) todisplay data•

create a pictograph and sample bar
graph of real objects and interpret
data by comparing quantities (forexample, more, fewer, less than,greater than)•create living graphs using real objects
and people*••

design a survey and systematicallycollect, record, organize and display
the data in a bar graph, circle graph,line graph•design a survey and systematicallycollect, organize and display data inpictographs and bar graphs•

collect, display and interpret data forthe purpose of answering questions•

create pictographs and tally marks•

express probabilities using scale (0–1)
or per cent (0%–100%).

•

When applying with understanding
learners:

set up a spreadsheet using simple
formulas to manipulate data and tocreate graphs•

When applying with understanding
learners:

use tree diagrams to express
probability using simple fractions.

•

When applying with understanding
learners:

express the chance of an event
happening using words or phrases(impossible, less likely, maybe, most
likely, certain).

•

identify the mode of a set of data•

When applying with understanding
learners:

represent the relationship between
objects in sets using tree, Venn andCarroll diagrams•

Learning continuums

Mathematics scope and sequence

Very young children view the world as aplace of possibilities. The teacher should
try to introduce practical examples andshould use appropriate vocabulary.
Discussions about chance in daily eventsshould be relevant to the context of thelearners.

Situations that come up naturally inthe classroom, often through literature,present opportunities for discussing
probability. Discussions need to takeplace in which students can share theirsense of likelihood in terms that are usefulto them.

Situations that come up naturally in theclassroom or form part of the units ofinquiry present opportunities for students
to further develop their understanding ofstatistics and probability concepts.

Technology also gives us the possibilityof rapidly replicating random events.
Computer and web-based applications
can be used to toss coins, roll dice, andtabulate and graph the results.

Technology gives us the option ofcreating a graph at the press of a key.
Being able to generate different typesof graphs allows learners to explore andappreciate the attributes of each type ofgraph and its efficacy in displaying thedata.

Learning continuums

Mathematics scope and sequence9

Learning continuums

Measurement
To measure is to attach a number to a quantity using a chosen unit. Since the attributes being measured arecontinuous, ways must be found to deal with quantities that fall between numbers. It is important to know
how accurate a measurement needs to be or can ever be.

Overall expectations
Phase 1
Learners will develop an understanding of how measurement involves the comparison of objects and theordering and sequencing of events. They will be able to identify, compare and describe attributes of real
objects as well as describe and sequence familiar events in their daily routine.

Phase 2
Learners will understand that standard units allow us to have a common language to measure anddescribe objects and events, and that while estimation is a strategy that can be applied for approximate
measurements, particular tools allow us to measure and describe attributes of objects and events with moreaccuracy. Learners will develop these understandings in relation to measurement involving length, mass,capacity, money, temperature and time.

Phase 3
Learners will continue to use standard units to measure objects, in particular developing their understandingof measuring perimeter, area and volume. They will select and use appropriate tools and units ofmeasurement, and will be able to describe measures that fall between two numbers on a scale. The learners
will be given the opportunity to construct meaning about the concept of an angle as a measure of rotation.

Phase 4
Learners will understand that a range of procedures exists to measure different attributes of objects andevents, for example, the use of formulas for finding area, perimeter and volume. They will be able to decideon the level of accuracy required for measuring and using decimal and fraction notation when precise
measurements are necessary. To demonstrate their understanding of angles as a measure of rotation, thelearners will be able to measure and construct angles.

10

Mathematics scope and sequenceMathematics scope and sequence

develop and describe formulas forfinding perimeter, area and volume
use decimal and fraction notation inmeasurement, for example, 3.2 cm,
1.47 kg, 1½ miles•

estimate and measure using standard
units of measurement: perimeter, areaand volume•

estimate and measure objects using
standard units of measurement:
length, mass, capacity, money andtemperature•

identify, compare and describe
attributes of real objects, for example,
longer, shorter, heavier, empty, full,hotter, colder•

understand an angle as a measure ofrotation.

••understand unit conversions within
measurement systems (metric orcustomary).

•

understand relationships between
units, for example, metres,
centimetres and millimetres•

When transferring meaning into
symbols learners:

understand the relationships between
area and perimeter, between area andvolume, and between volume andcapacity•

understand that measures can fall
between numbers on a measurement
scale, for example, 3½ kg, between
4 cm and 5 cm•

When transferring meaning into
symbols learners:

understand procedures for findingarea, perimeter and volume•

understand the use of standard
units to measure perimeter, area andvolume•When transferring meaning into
symbols learners:

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:

When transferring meaning into
symbols learners:

understand that time is measured
using universal units of measure, forexample, years, months, days, hours,
minutes and seconds.

understand that events in daily
routines can be described andsequenced, for example, before, after,
bedtime, storytime, today, tomorrow.

••understand that tools can be used tomeasure•understand that calendars can beused to determine the date, and to
identify and sequence days of theweek and months of the year

understand the use of standard units
to measure, for example, length, mass,money, time, temperature•

understand that attributes of real
objects can be compared anddescribed, for example, longer,
shorter, heavier, empty, full, hotter,colder••Learning outcomes
When constructing meaning learners:

Estimation allows us to measure withdifferent levels of accuracy.

Learning outcomes
When constructing meaning learners:

Events can be ordered and sequenced.

A range of procedures exists to measuredifferent attributes of objects and events.

Conversion of units and measurementsallows us to make sense of the world welive in.

Relationships exist between standard
units that measure the same attributes.

We use tools to measure the attributes ofobjects and events.

Conceptual understandings
Accuracy of measurements depends on
the situation and the precision of the tool.

Conceptual understandings
Objects and events have attributes thatcan be measured using appropriate tools.

Conceptual understandings
Standard units allow us to have a common
language to identify, compare, order andsequence objects and events.

Conceptual understandings
Measurement involves comparing objects
and events.

Objects have attributes that can bemeasured using non-standard units.

Phase 4Phase 3Phase 2Phase 1

Learning continuum for measurement

Learning continuums

11

12

select and use appropriate unitsof measurement and tools to solve
problems in real-life situations
determine and justify the level ofaccuracy required to solve real-life
problems involving measurementuse decimal and fractional notation
in measurement, for example, 3.2 cm,
1.47 kg, 1½ miles
use timetables and schedules (12hour and 24-hour clocks) in real-lifesituations
determine times worldwide.

•

•

•

•

•

use standard units of measurement tosolve problems in real-life situations
involving perimeter, area and volume
select appropriate tools and units ofmeasurement
use timelines in units of inquiry andother real-life situations.

•••use standard units of measurement tosolve problems in real-life situations
involving length, mass, capacity,money and temperature
use measures of time to assist withproblem solving in real-life situations.

••describe observations about eventsand objects in real-life situations

use non-standard units ofmeasurement to solve problems inreal-life situations involving length,
mass and capacity.

••

carry out simple unit conversions
within a system of measurement(metric or customary).

•

When applying with understanding
learners:

measure and construct angles indegrees using a protractor••

read and write digital and analogue
time on 12-hour and 24-hour clocks.

•

read and interpret scales on a range ofmeasuring instruments•

When applying with understanding
learners:

estimate and compare lengths of time:
second, minute, hour, day, week andmonth.

•

describe measures that fall between
numbers on a scale•

When applying with understanding
learners:

identify, describe and sequence
events in their daily routine, forexample, before, after, bedtime,
storytime, today, tomorrow.

•read and write the time to the hour,half hour and quarter hour•

When applying with understanding
learners:

compare the length, mass andcapacity of objects using nonstandard units•

Learning continuums

Mathematics scope and sequenceNotesLearners need many opportunities toexperience and quantify measurement
in a direct kinesthetic manner. They willdevelop understanding of measurement
by using manipulatives and materials
from their immediate environment, forexample, containers of different sizes,
sand, water, beads, corks and beans.

Notes
Using materials from their immediate
environment, learners can investigate
how units are used for measurement
and how measurements vary dependingon the unit that is used. Learners willrefine their estimation and measurement
skills by basing estimations on priorknowledge, measuring the object andcomparing actual measurements with
their estimations.

Mathematics scope and sequence

Please note that outcomes relating toangles also appear in the shape and spacestrand.

A wide range of measuring tools should
be available to the students, for example,
rulers, trundle wheels, tape measures,
bathroom scales, kitchen scales,
timers, analogue clocks, digital clocks,
stopwatches and calendars. There are an
increasing number of computer and webbased applications available for students
to use in authentic contexts.

Notes
In order to use measurement moreauthentically, learners should have theopportunity to measure real objects inreal situations. The units of inquiry canoften provide these realistic contexts.
While the emphasis for understanding ison measurement systems commonly usedin the learner’s world, it is worthwhile
being aware of the existence of othersystems and how conversions between
systems help us to make sense of them.

Notes
Learners generalize their measuringexperiences as they devise proceduresand formulas for working out perimeter,area and volume.

Learning continuums

13Learning continuums

Shape and space
The regions, paths and boundaries of natural space can be described by shape. An understanding of theinterrelationships of shape allows us to interpret, understand and appreciate our two-dimensional (2D) andthree-dimensional (3D) world.

Overall expectations
Phase 1
Learners will understand that shapes have characteristics that can be described and compared. They will
understand and use common language to describe paths, regions and boundaries of their immediate
environment.

Phase 2
Learners will continue to work with 2D and 3D shapes, developing the understanding that shapes areclassified and named according to their properties. They will understand that examples of symmetry andtransformations can be found in their immediate environment. Learners will interpret, create and use simple
directions and specific vocabulary to describe paths, regions, positions and boundaries of their immediate
environment.

Phase 3
Learners will sort, describe and model regular and irregular polygons, developing an understanding of
their properties. They will be able to describe and model congruency and similarity in 2D shapes. Learners
will continue to develop their understanding of symmetry, in particular reflective and rotational symmetry.
They will understand how geometric shapes and associated vocabulary are useful for representing anddescribing objects and events in real-world situations.

Phase 4
Learners will understand the properties of regular and irregular polyhedra. They will understand theproperties of 2D shapes and understand that 2D representations of 3D objects can be used to visualize
and solve problems in the real world, for example, through the use of drawing and modelling. Learners willdevelop their understanding of the use of scale (ratio) to enlarge and reduce shapes. They will apply thelanguage and notation of bearing to describe direction and position.

14

Mathematics scope and sequenceMathematics scope and sequence

understand the properties of regularand irregular polyhedra
understand the properties of circles
understand how scale (ratios) is used
to enlarge and reduce shapes
understand systems for describing
position and direction

•
•
•
•

understand the properties of regularand irregular polygons
understand congruent or similar
shapes••understand that 2D and 3D shapes
can be created by putting together
and/or taking apart other shapes
understand that examples ofsymmetry and transformations can befound in their immediate environment
understand that geometric shapes
are useful for representing real-world
situations
understand that directions can beused to describe pathways, regions,
positions and boundaries of their
immediate environment.

•

•

•

•

understand that common language
can be used to describe position anddirection, for example, inside, outside,
above, below, next to, behind, in frontof, up, down.

•

understand an angle as a measure ofrotationunderstand that directions forlocation can be represented bycoordinates on a gridunderstand that visualization of shapeand space is a strategy for solving
problems.

•••

understand that geometric ideas
and relationships can be used tosolve problems in other areas ofmathematics and in real life.

understand the common language
used to describe shapes•

understand the common language
used to describe shapes•

understand that there are
relationships among and between 2D
and 3D shapes•

understand that 2D and 3D shapes
have characteristics that can be
described and compared••

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:

understand that 2D representationsof 3D objects can be used to visualize
and solve problems

Geometric tools and methods can beused to solve problems relating to shapeand space.

Geometric shapes and vocabulary areuseful for representing and describing
objects and events in real-world
situations.

Specific vocabulary can be used todescribe an object’s position in space.

•

Consolidating what we know ofgeometric concepts allows us to makesense of and interact with our world.

Shapes can be transformed in different
ways.

Some shapes are made up of parts that
repeat in some way.

Objects in our immediate environment
have a position in space that can be
described according to a point ofreference.

understand that lines and axes ofreflective and rotational symmetry
assist with the construction of shapes

Conceptual understandings
Manipulation of shape and space takes
place for a particular purpose.

Conceptual understandings
Changing the position of a shape does
not alter its properties.

Conceptual understandings
Shapes are classified and named
according to their properties.

Conceptual understandings
Shapes can be described and organizedaccording to their properties.

•

Phase 4Phase 3Phase 2Phase 1

Learning continuum for shape and space

Learning continuums

15

16

use geometric vocabulary whendescribing shape and space inmathematical situations and beyond
use scale (ratios) to enlarge andreduce shapes
apply the language and notation of
bearing to describe direction andpositionuse 2D representations of 3D objects
to visualize and solve problems, forexample using drawings or models.

•

•
•

•

analyse and describe 2D and 3D
shapes, including regular andirregular polygons, using geometrical
vocabulary
identify, describe and model
congruency and similarity in 2D
shapesrecognize and explain symmetricalpatterns, including tessellation, in theenvironment
apply knowledge of transformations
to problem-solving situations.

•

•

•

•

recognize and explain simple
symmetrical designs in theenvironment
apply knowledge of symmetry to
problem-solving situations
interpret and use simple directions,describing paths, regions, positions
and boundaries of their immediate
environment.

•••

explore and describe the paths,
regions and boundaries of their
immediate environment (inside,outside, above, below) and theirposition (next to, behind, in front of,up, down).

analyse and use what they know
about 3D shapes to describe and workwith 2D shapes

explore the use of geometric ideas
and relationships to solve problems inother areas of mathematics.

•

interpret and create simple directions,describing paths, regions, positions
and boundaries of their immediate
environment.

••

create and model how a 2D net
converts into a 3D shape and vice
versa

•

describe and/or represent mental
images of objects, patterns, andpaths.

•

represent ideas about the real worldusing geometric vocabulary andsymbols, for example, through oral
description, drawing, modelling,labelling••

identify and use the language andnotation of bearing to describe
direction and position•

locate features on a grid using
coordinates•

identify lines of reflective symmetry•

When applying with understanding
learners:

identify and use scale (ratios) toenlarge and reduce shapes•

analyse angles by comparing anddescribing rotations: whole turn; halfturn; quarter turn; north, south, east
and west on a compass•

create and describe symmetrical andtessellating patterns•

When applying with understanding
learners:

describe lines and angles using
geometric vocabulary•

describe and model congruency andsimilarity in 2D shapes•

analyse and describe the relationships
between 2D and 3D shapes•

describe position and direction, forexample, inside, outside, above,
below, next to, behind, in front of, up,
down.

•

When applying with understanding
learners:

analyse, describe, classify and visualize
2D (including circles, triangles andquadrilaterals) and 3D shapes, using
geometric vocabulary•sort, describe and model regular andirregular polygons•

sort, describe and label 2D and 3D
shapes•sort, describe and compare 3D shapes•

When applying with understanding
learners:

When transferring meaning into
symbols learners:

When transferring meaning into
symbols learners:

When transferring meaning into
symbols learners:

When transferring meaning into
symbols learners:

Learning continuums

Mathematics scope and sequenceLearners need to understand theproperties of 2D and 3D shapes
before the mathematical vocabulary
associated with shapes makes senseto them. Through creating andmanipulating shapes, learners align theirnatural vocabulary with more formal
mathematical vocabulary and begin toappreciate the need for this precision.

Learners need many opportunities toexperience shape and space in a directkinesthetic manner, for example, throughplay, construction and movement.

The manipulatives that they interact withshould include a range of 3D shapes, inparticular the real-life objects with whichchildren are familiar. 2D shapes (plane
shapes) are a more abstract concept but
can be understood as faces of 3D shapes.

Notes

Notes
Tools such as compasses and protractors
are commonly used to solve problems inreal-life situations. However, care should
be taken to ensure that students have astrong understanding of the concepts
embedded in the problem to ensure
meaningful engagement with the tools
and full understanding of the solution.

Computer and web-based applications
can be used to explore shape and spaceconcepts such as symmetry, angles andcoordinates.
The units of inquiry can provide authentic
contexts for developing understandingof concepts relating to location anddirections.

Notes

Notes

Learning continuums

Mathematics scope and sequence17

Learning continuums

Pattern and function
To identify pattern is to begin to understand how mathematics applies to the world in which we live. The
repetitive features of patterns can be identified and described as generalized rules called “functions”. Thisbuilds a foundation for the later study of algebra.

Overall expectations
Phase 1
Learners will understand that patterns and sequences occur in everyday situations. They will be able toidentify, describe, extend and create patterns in various ways.

Phase 2
Learners will understand that whole numbers exhibit patterns and relationships that can be observed
and described, and that the patterns can be represented using numbers and other symbols. As a result,learners will understand the inverse relationship between addition and subtraction, and the associative andcommutative properties of addition. They will be able to use their understanding of pattern to represent
and make sense of real-life situations and, where appropriate, to solve problems involving addition andsubtraction.

Phase 3
Learners will analyse patterns and identify rules for patterns, developing the understanding that functions
describe the relationship or rules that uniquely associate members of one set with members of anotherset. They will understand the inverse relationship between multiplication and division, and the associativeand commutative properties of multiplication. They will be able to use their understanding of pattern andfunction to represent and make sense of real-life situations and, where appropriate, to solve problems
involving the four operations.

Phase 4
Learners will understand that patterns can be represented, analysed and generalized using algebraic
expressions, equations or functions. They will use words, tables, graphs and, where possible, symbolic rules
to analyse and represent patterns. They will develop an understanding of exponential notation as a way toexpress repeated products, and of the inverse relationship that exists between exponents and roots. The
students will continue to use their understanding of pattern and function to represent and make sense ofreal-life situations and to solve problems involving the four operations.

18

Mathematics scope and sequence

Conceptual understandings
Patterns can often be generalized using
algebraic expressions, equations orfunctions.
Exponential notation is a powerful way
to express repeated products of the same
number.
Learning outcomes
When constructing meaning learners:
understand that patterns can begeneralized by a rule
understand exponents as repeatedmultiplication
understand the inverse relationshipbetween exponents and roots
understand that patterns can be
represented, analysed and generalizedusing tables, graphs, words, and,
when possible, symbolic rules.

•
•
•
•

Conceptual understandings
Functions are relationships or rules thatuniquely associate members of one set
with members of another set.
By analysing patterns and identifying
rules for patterns it is possible to makepredictions.
Learning outcomes
When constructing meaning learners:
understand that patterns can beanalysed and rules identified
understand that multiplication isrepeated addition and that division isrepeated subtraction
understand the inverse relationshipbetween multiplication and division
understand the associativeand commutative properties ofmultiplication.

•
•

•
•

Conceptual understandings
Whole numbers exhibit patterns andrelationships that can be observed anddescribed.
Patterns can be represented using
numbers and other symbols.
Learning outcomes
When constructing meaning learners:
understand that patterns can befound in numbers, for example, odd
and even numbers, skip counting
understand the inverse relationshipbetween addition and subtraction
understand the associative andcommutative properties of addition.

•••

Conceptual understandings
Patterns and sequences occur in everyday
situations.

Learning outcomes
When constructing meaning learners:•

Mathematics scope and sequence

represent the rule of a pattern byusing a function
analyse pattern and function using
words, tables and graphs, and, whenpossible, symbolic rules.

••

describe the rule for a pattern in avariety of ways
represent rules for patterns using
words, symbols and tables
identify a sequence of operations
relating one set of numbers to another
set.

•
•
•

represent patterns in a variety of ways,
for example, using words, drawings,
symbols, materials, actions, numbers
describe number patterns, forexample, odd and even numbers, skip
counting.

•••

describe patterns in various ways,
for example, using words, drawings,
symbols, materials, actions, numbers.

When transferring meaning into
symbols learners:

When transferring meaning into
symbols learners:

When transferring meaning into
symbols learners:

When transferring meaning into
symbols learners:

understand that patterns can befound in everyday situations, forexample, sounds, actions, objects,
nature.

Patterns repeat and grow.

Phase 4Phase 3Phase 2Phase 1

Learning continuum for pattern and function

Learning continuums

19

20

use number patterns to represent andunderstand real-life situations
use the properties and relationshipsof addition and subtraction to solve
problems.

••

A range of manipulatives can be used to
explore patterns including pattern blocks,
attribute blocks, colour tiles, calculators,
number charts, beans and buttons.

Notes
The world is filled with pattern and there
will be many opportunities for learners
to make this connection across thecurriculum.
Four-function calculators can be used to
explore number patterns.

Notes
Students will apply their understandingof pattern to the numbers they alreadyknow. The patterns they find will help todeepen their understanding of a range of
number concepts.

extend and create patterns in
numbers, for example, odd and evennumbers, skip counting••

extend and create patterns.

When applying with understanding
learners:

When applying with understanding
learners:

use functions to solve problems.

•

use number patterns to makepredictions and solve problems
use the properties and relationships ofthe four operations to solve problems.

•
•

Notes
Algebra is a mathematical language
using numbers and symbols to express
relationships. When the same relationshipworks with any number, algebra uses
letters to represent the generalization.
Letters can be used to represent thequantity.

select appropriate methods to analyse
patterns and identify rules•

select appropriate methods forrepresenting patterns, for example
using words, symbols and tables

•

Notes
Patterns are central to the understandingof all concepts in mathematics. They are
the basis of how our number system isorganized. Searching for, and identifying,patterns helps us to see relationships,
make generalizations, and is a powerfulstrategy for problem solving. Functions
develop from the study of patterns andmake it possible to predict in mathematics
problems.

When applying with understanding
learners:

When applying with understanding
learners:

Learning continuums

Mathematics scope and sequenceLearning continuums

Number
Our number system is a language for describing quantities and the relationships between quantities. For
example, the value attributed to a digit depends on its place within a base system.
Numbers are used to interpret information, make decisions and solve problems. For example, the operationsof addition, subtraction, multiplication and division are related to one another and are used to process
information in order to solve problems. The degree of precision needed in calculating depends on how theresult will be used.

Overall expectations
Phase 1
Learners will understand that numbers are used for many different purposes in the real world. They willdevelop an understanding of one-to-one correspondence and conservation of number, and be able tocount and use number words and numerals to represent quantities.

Phase 2
Learners will develop their understanding of the base 10 place value system and will model, read, write,
estimate, compare and order numbers to hundreds or beyond. They will have automatic recall of additionand subtraction facts and be able to model addition and subtraction of whole numbers using the appropriate
mathematical language to describe their mental and written strategies. Learners will have an understandingof fractions as representations of whole-part relationships and will be able to model fractions and use
fraction names in real-life situations.

Phase 3
Learners will develop the understanding that fractions and decimals are ways of representing whole-part
relationships and will demonstrate this understanding by modelling equivalent fractions and decimal
fractions to hundredths or beyond. They will be able to model, read, write, compare and order fractions, anduse them in real-life situations. Learners will have automatic recall of addition, subtraction, multiplicationand division facts. They will select, use and describe a range of strategies to solve problems involving
addition, subtraction, multiplication and division, using estimation strategies to check the reasonablenessof their answers.

Phase 4
Learners will understand that the base 10 place value system extends infinitely in two directions and will
be able to model, compare, read, write and order numbers to millions or beyond, as well as model integers.
They will develop an understanding of ratios. They will understand that fractions, decimals and percentages
are ways of representing whole-part relationships and will work towards modelling, comparing, reading,writing, ordering and converting fractions, decimals and percentages. They will use mental and written
strategies to solve problems involving whole numbers, fractions and decimals in real-life situations, using a
range of strategies to evaluate reasonableness of answers.

Mathematics scope and sequence

21

22

The operations of addition, subtraction,
multiplication and division are related
to each other and are used to process
information to solve problems.

Numbers are connected to each otherthrough a variety of relationships.

Making connections between our
experiences with number can help us todevelop number sense.

estimate sums and differences

recognize groups of zero to five
objects without counting (subitizing)••understand the relative magnitude ofwhole numbers•

use the language of multiplicationand division, for example, factor,
multiple, product, quotient, prime
numbers, composite number•

develop strategies for memorizing
addition and subtraction number
facts

•

understand conservation of number*•model multiplication and division ofwhole numbers•

model addition and subtraction ofwhole numbers•

simplify fractions using manipulatives
model decimal fractions tothousandths or beyond
model percentages
understand the relationship between
fractions, decimals and percentages

•
•
•
•

model improper fractions and mixed
numbers

model exponents and square roots••

use the language of addition andsubtraction, for example, add, takeaway, plus, minus, sum, difference•

understand that numbers canbe constructed in multiple ways,
for example, by combining andpartitioning•

model decimal fractions tohundredths or beyond

use the language of fractions, forexample, numerator, denominator•model simple fraction relationships••

model ratios•

model equivalent fractions•

estimate quantities to 100 or beyond•

understand that, for a set of objects,
the number name of the last object
counted describes the quantity of thewhole set•model integers in appropriate
contexts

model numbers to millions or beyond
using the base 10 place value system•model numbers to thousands or
beyond using the base 10 place value
system•model numbers to hundreds or
beyond using the base 10 place value
system**•understand one-to-one
correspondence••

Learning outcomes
When constructing meaning learners:

Learning outcomes
When constructing meaning learners:

Ratios are a comparison of two numbers
or quantities.

Learning outcomes
When constructing meaning learners:

Even complex operations can bemodelled in a variety of ways, forexample, an algorithm is a way torepresent an operation.

Learning outcomes
When constructing meaning learners:

There are many mental methods that canbe applied for exact and approximate
computations.

Number operations can be modelled in avariety of ways.

Fractions, decimal fractions andpercentages are ways of representing
whole-part relationships.

Fractions and decimals are ways ofrepresenting whole-part relationships.

Fractions are ways of representing wholepart relationships.

Numbers can be used in many ways fordifferent purposes in the real world.

For fractional and decimal computation,
the ideas developed for whole-number
computation can apply.

Conceptual understandings
The base 10 place value system extends
infinitely in two directions.

Conceptual understandings
The base 10 place value system can beextended to represent magnitude.

Conceptual understandings
The base 10 place value system is used
to represent numbers and number
relationships.

Conceptual understandings
Numbers are a naming system.

The operations of addition, subtraction,
multiplication and division are related
to each other and are used to process
information to solve problems.

Phase 4Phase 3Phase 2Phase 1

Learning continuum for number

Learning continuums

Mathematics scope and sequenceMathematics scope and sequence

read and write integers in appropriate
contexts•

read, write, compare and order
fractionsread and write equivalent fractions
read, write, compare and order
fractions to hundredths or beyond
describe mental and written
strategies for multiplication anddivision.

••••

describe mental and written
strategies for adding and subtracting
two-digit numbers.

•

use whole numbers up to millions or
beyond in real-life situations
use ratios in real-life situations
use integers in real-life situations•••

use whole numbers up to thousands
or beyond in real-life situations
use fast recall of multiplication anddivision number facts in real-lifesituations••

use whole numbers up to hundreds or
beyond in real-life situations
use cardinal and ordinal numbers inreal-life situations••

count to determine the number ofobjects in a set

use number words and numerals
to represent quantities in real-lifesituations••

convert between fractions, decimals
and percentages.

•

When applying with understanding
learners:

read, write, compare and order
percentages•

When applying with understanding
learners:

read, write, compare and orderdecimal fractions to thousandths or
beyond•When applying with understanding
learners:

simplify fractions in mental andwritten form•

When applying with understanding
learners:

convert improper fractions to mixed
numbers and vice versa•

read and write exponents and squareroots

read and write ratios•

develop strategies for memorizing
addition, subtraction, multiplicationand division number facts•

read, write, compare and order
cardinal and ordinal numbers••

read, write, compare and order whole
numbers up to millions or beyond•

read, write, compare and order whole
numbers up to thousands or beyond•

read and write whole numbers up to
hundreds or beyond

connect number names andnumerals to the quantities they
represent.

•

model addition, subtraction,
multiplication and division ofdecimals.

••model addition and subtraction ofdecimals.

•model addition, subtraction,
multiplication and division of fractions•

When transferring meaning into
symbols learners:

model addition and subtraction of
fractions with the same denominator.

•

model addition and subtractionof fractions with relateddenominators***•

When transferring meaning into
symbols learners:

understand situations that involve
multiplication and division•

When transferring meaning into
symbols learners:

use the language of mathematics
to compare quantities, for example,
more, less, first, second.

understand whole-part relationships

When transferring meaning into
symbols learners:••

Learning continuums

23use the language of mathematics
to compare quantities in real-lifesituations, for example, more, less,first, second

subitize in real-life situations

use simple fraction names in real-lifesituations.

•

•

24

•

use mental and written strategies for
addition and subtraction of twodigit numbers or beyond in real-lifesituations
select an appropriate method forsolving a problem, for example,
mental estimation, mental or written
strategies, or by using a calculator
use strategies to evaluate thereasonableness of answers.

••

use fractions in real-life situations••

use fast recall of addition andsubtraction number facts in real-lifesituations•

use decimal fractions in real-lifesituations
use mental and written strategies for
multiplication and division in real-lifesituations
select an efficient method forsolving a problem, for example,
mental estimation, mental or written
strategies, or by using a calculator
use strategies to evaluate thereasonableness of answers
add and subtract fractions withrelated denominators in real-lifesituations
add and subtract decimals in real-lifesituations, including money
estimate sum, difference, productand quotient in real-life situations,
including fractions and decimals.

•
•

•

•
•

•
•

convert improper fractions to mixed
numbers and vice versa in real-lifesituations
simplify fractions in computation
answers
use fractions, decimals andpercentages interchangeably in reallife situations
select and use an appropriate
sequence of operations to solve word
problems
select an efficient method for solving
a problem: mental estimation, mental
computation, written algorithms, byusing a calculator
use strategies to evaluate thereasonableness of answers
use mental and written strategies foradding, subtracting, multiplying anddividing fractions and decimals in
real-life situations
estimate and make approximations inreal-life situations involving fractions,decimals and percentages.

•

•
•

•

•

•
•

•

Learning continuums

Mathematics scope and sequenceMathematics scope and sequence

There are many opportunities in the units
of inquiry and during the school day forstudents to practise and apply number
concepts authentically.

Estimation is a skill that will develop
with experience and will help children
gain a “feel” for numbers. Children mustbe given the opportunity to check theirestimates so that they are able to further
refine and improve their estimation skills.

Learners who have been encouraged
to select their own apparatus andmethods, and who become accustomed
to discussing and questioning theirwork, will have confidence in looking foralternative approaches when an initialattempt is unsuccessful.

Notes*To conserve, in mathematical terms,
means the amount stays the same
regardless of the arrangement.

Difficulties with fractions can arise whenfractional notation is introduced before
students have fully constructed meaning
about fraction concepts.

To be useful, addition and subtraction
facts need to be recalled automatically.
Research clearly indicates that there are
more effective ways to do this than “drill
and practice”. Above all, it helps to havestrategies for working them out. Counting
on, using doubles and using 10s are goodstrategies, although learners frequently
invent methods that work equally well forthemselves.

Students need to use numbers inmany situations in order to apply theirunderstanding to new situations. In
addition to the units of inquiry, children’s
literature also provides rich opportunities
for developing number concepts.

Notes**Modelling involves using concretematerials to represent numbers or
number operations, for example, the useof pattern blocks or fraction pieces torepresent fractions and the use of base 10
blocks to represent number operations.

By reflecting on and recording theirfindings in mathematics learning logs,
students begin to notice patterns in thenumbers that will further develop theirunderstanding.

Calculator skills must not be ignored.
All answers should be checked for theirreasonableness.

The interpretation and meaning of
remainders can cause difficulty forsome learners. This is especially true if
calculators are being used. For example,
67 ÷ 4 = 16.75. This can also be shownas 16¾ or 16 r3. Learners need practicein producing appropriate answers whenusing remainders. For example, for aschool trip with 25 students, only buses
that carry 20 students are available. A
remainder could not be left behind, so
another bus would be required!

***Examples of related denominators
include halves, quarters (fourths) andeighths. These can be modelled easily byfolding strips or squares of paper.

Notes
Modelling using manipulatives provides
a valuable scaffold for constructing
meaning about mathematicalconcepts. There should be regular
opportunities for learners to workwith a range of manipulatives and to
discuss and negotiate their developing
understandings with others.

A thorough understanding ofmultiplication, factors and large
numbers is required before working withexponents.

Students should be given many
opportunities to discover the link
between fractions and division.

Measurement is an excellent way of
exploring the use of fractions anddecimals and their interchange.

Estimation plays a key role in checking
the feasibility of answers. The methodof multiplying numbers and ignoring
the decimal point, then adjusting theanswer by counting decimal places, does
not give the learner an understanding of
why it is done. Application of place value
knowledge must precede this applicationof pattern.

Notes
It is not practical to continue to develop
and use base 10 materials beyond 1,000.
Learners should have little difficulty inextending the place value system once
they have understood the grouping
pattern up to 1,000. There are a number
of websites where virtual manipulatives
can be utilized for working with larger
numbers.

Learning continuums

25SamplesSeveral examples of how schools are using the planner to facilitate mathematics inquiries have beendeveloped and trialled by IB World Schools offering the PYP. These examples are included in the HTMLversion of the mathematics scope and sequence on the online curriculum centre. The IB is interested
in receiving planners that have been developed for mathematics inquiries or for units of inquiry wheremathematics concepts are strongly evident. Please send planners to pyp@ibo.org for possible inclusion on
this site.

26

Mathematics scope and sequence