Mathematics scope and sequence
Primary Years Programme
Mathematics scope and sequence
Primary Years Programme
PYP110Printed in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire
Published February 2009
International BaccalaureatePeterson House, Malthouse Avenue, Cardiff Gate
Cardiff, Wales GB CF23 8GLUnited Kingdom
Phone: +44 29 2054 7777Fax: +44 29 2054 7778
Website: http://www.ibo.org
© International Baccalaureate Organization 2009
The International Baccalaureate (IB) offers three high quality and challenging educational programmes for a worldwide community of schools, aiming to create a better, more peaceful world.
The IB is grateful for permission to reproduce and/or translate any copyright material used in this publication. Acknowledgments are included, where appropriate, and, if notified, the IB will be pleased to rectify any errors or omissions at the earliest opportunity.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written permission of the IB, or as expressly permitted by law or by the IB’s own rules and policy. See http://www.ibo.org/copyright.
IB merchandise and publications can be purchased through the IB store at http://store.ibo.org. General ordering queries should be directed to the sales and marketing department in Cardiff.
Phone: +44 29 2054 7746Fax: +44 29 2054 7779Email: [email protected]
Primary Years Programme
Mathematics scope and sequence
IB mission statementThe International Baccalaureate aims to develop inquiring, knowledgeable and caring young people who help to create 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 profileThe 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 this love of learning will be sustained throughout their lives.
Knowledgeable They explore concepts, ideas and issues that have local and global significance. In so doing, they acquire in-depth knowledge and develop understanding across a broad and balanced 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 more than 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 are accustomed to seeking and evaluating a range of points of view, and are willing to grow from the experience.
Caring They 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 the lives 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 are brave and articulate in defending their beliefs.
Balanced They understand the importance of intellectual, physical and emotional balance to achieve 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 learning and personal development.
© International Baccalaureate Organization 2007
Mathematics scope and sequence
Contents
Introduction to the PYP mathematics scope and sequence 1
What the PYP believes about learning mathematics 1
Mathematics in a transdisciplinary programme 2
The structure of the PYP mathematics scope and sequence 3
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 1
Introduction to the PYP mathematics scope and 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 mathematicsThe power of mathematics for describing and analysing the world around us is such that it has become a highly 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 that students 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 that students 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 mathematicsIt 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 than by 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).
Applying with understanding
Transferring meaning
Constructing meaning
Figure 1How children learn mathematics
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence2
Constructing meaning about mathematicsLearners 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 symbolsOnly 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 to describe their understanding using their own method of symbolic notation, then learning to transfer them into conventional mathematical notation.
Applying with understandingApplying with understanding can be viewed as the learners demonstrating and acting on their understanding. Through authentic activities, learners should independently select and use appropriate symbolic notation to process and record their thinking. These authentic activities should include a range of practical hands-on problem-solving activities and realistic situations that provide the opportunity to demonstrate mathematical thinking through presented or recorded formats. In this way, learners are able to apply 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 mathematical reasoning.
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 mathematical situations. 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 this interactive process.
Mathematics in a transdisciplinary programmeWherever possible, mathematics should be taught through the relevant, realistic context of the units of inquiry. The direct teaching of mathematics in a unit of inquiry may not always be feasible but, where appropriate, 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 “big ideas” within and between the different strands of mathematics, the programme of inquiry and other subject areas.
Links to the transdisciplinary themes should be explicitly made, whether or not the mathematics is being taught within the programme of inquiry. A developing understanding of these links will contribute to the students’ understanding of mathematics in the world and to their understanding of the transdisciplinary
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence 3
theme. The role of inquiry in mathematics is important, regardless of whether it is being taught inside or outside the programme of inquiry. However, it should also be recognized that there are occasions when it is preferable for students to be given a series of strategies for learning mathematical skills in order to progress in their mathematical understanding rather than struggling to proceed.
The structure of the PYP mathematics scope and sequenceThis scope and sequence aims to provide information for the whole school community of the learning that is going 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 phase building 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 described in the behaviours or learning outcomes associated with each phase and these learning outcomes relate specifically 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 of mathematical 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 and represent this data in graphs. In this type of example, perhaps the difference between the two stages is that 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 and behaviours that students might demonstrate when applying with understanding. It is important to note that 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 and other 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 the phases. 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
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence4
meaning into symbols and applying with understanding stages of the same phase. However, on some occasions, a mathematical concept is introduced in one phase but students are not expected to apply the concept until a later phase. This is a deliberate decision aimed at providing students with adequate time and opportunities 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 and sequenceIn 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 and accompanying continuums and how they can be used to inform planning, teaching and assessing of mathematics 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 these continuums.
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 the needs of their students and their particular teaching situations.
The continuums are not prescriptive tools that assume a learner must attain all the outcomes of a particular phase before moving on to the next phase, nor that the learner should be in the same phase for each strand.
Each teacher needs to identify the extent to which these factors affect the learner. Plotting a mathematical profile for each student is a complex process for PYP teachers. Prior knowledge should therefore never be assumed 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 scope and 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 be read and considered.
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence 5
Viewing a unit of inquiry through the lens of mathematicsThe 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 and knowledge 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 subject-specific 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.
Figure 2Sample processes for viewing a unit of inquiry through the lens of mathematics
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).
Mathematics scope and sequence6
Learning continuums
Data handlingData 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 highlight similarities, 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 expectationsPhase 1Learners will develop an understanding of how the collection and organization of information helps to make sense of the world. They will sort, describe and label objects by attributes and represent information in graphs including pictographs and tally marks. The learners will discuss chance in daily events.
Phase 2Learners will understand how information can be expressed as organized and structured data and that this can occur in a range of ways. They will collect and represent data in different types of graphs, interpreting the resulting information for the purpose of answering questions. The learners will develop an understanding that some events in daily life are more likely to happen than others and they will identify and describe likelihood using appropriate vocabulary.
Phase 3Learners 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 4Learners 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 and manipulate 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 (0–1 or 0%–100%) and that the probability of an event can be predicted theoretically.
Learning continuums
Mathem
atics scope and sequence7
Learning continuum for data handling
Phase 1 Phase 2 Phase 3 Phase 4
Conceptual understandingsWe collect information to make sense of the world around us.
Organizing objects and events helps us to solve problems.
Events in daily life involve chance.
Conceptual understandingsInformation can be expressed as organized and structured data.
Objects and events can be organized in different ways.
Some events in daily life are more likely to happen than others.
Conceptual understandingsData can be collected, organized, displayed and analysed in different ways.
Different graph forms highlight different aspects of data more efficiently.
Probability can be based on experimental events in daily life.
Probability can be expressed in numerical notations.
Conceptual understandingsData can be presented effectively for valid interpretation and communication.
Range, mode, median and mean can be used to analyse statistical data.
Probability can be represented on a scale between 0–1 or 0%–100%.
The probability of an event can be predicted theoretically.
Learning outcomesWhen constructing meaning learners:
understand that sets can be organized by different attributes
understand that information about themselves and their surroundings can be obtained in different ways
discuss chance in daily events (impossible, maybe, certain).
Learning outcomesWhen constructing meaning learners:
understand that sets can be organized by one or more attributes
understand that information about themselves and their surroundings can be collected and recorded in different ways
understand the concept of chance in daily events (impossible, less likely, maybe, most likely, certain).
Learning outcomesWhen constructing meaning learners:
understand that data can be collected, displayed and interpreted using simple graphs, for example, bar graphs, line graphs
understand that scale can represent different quantities in graphs
understand that the mode can be used to summarize a set of data
understand that one of the purposes of a database is to answer questions and solve problems
understand that probability is based on experimental events.
Learning outcomesWhen constructing meaning learners:
understand that different types of graphs have special purposes
understand that the mode, median, mean and range can summarize a set of data
understand that probability can be expressed in scale (0–1) or per cent (0%–100%)
understand the difference between experimental and theoretical probability.
When transferring meaning into symbols learners:
represent information through pictographs and tally marks
sort and label real objects by attributes.
When transferring meaning into symbols learners:
collect and represent data in different types of graphs, for example, tally marks, bar graphs
When transferring meaning into symbols learners:
collect, display and interpret data using simple graphs, for example, bar graphs, line graphs
identify, read and interpret range and scale on graphs
When transferring meaning into symbols learners:
collect, display and interpret data in circle graphs (pie charts) and line graphs
identify, describe and explain the range, mode, median and mean in a set of data
Learning continuums
Mathem
atics scope and sequence8
represent the relationship between objects in sets using tree, Venn and Carroll diagrams
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
use tree diagrams to express probability using simple fractions.
set up a spreadsheet using simple formulas to manipulate data and to create graphs
express probabilities using scale (0–1) or per cent (0%–100%).
When applying with understanding learners:
create pictographs and tally marks
create living graphs using real objects and people*
describe real objects and events by attributes.
When applying with understanding learners:
collect, display and interpret data for the purpose of answering questions
create a pictograph and sample bar graph of real objects and interpret data by comparing quantities (for example, more, fewer, less than, greater than)
use tree, Venn and Carroll diagrams to explore relationships between data
identify and describe chance in daily events (impossible, less likely, maybe, most likely, certain).
When applying with understanding learners:
design a survey and systematically collect, organize and display data in pictographs and bar graphs
select appropriate graph form(s) to display data
interpret range and scale on graphs
use probability to determine mathematically fair and unfair games and to explain possible outcomes
express probability using simple fractions.
When applying with understanding learners:
design a survey and systematically collect, record, organize and display the data in a bar graph, circle graph, line graph
identify, describe and explain the range, mode, median and mean in a set of data
create and manipulate an electronic database for their own purposes
determine the theoretical probability of an event and explain why it might differ from experimental probability.
NotesUnits of inquiry will be rich in opportunities for collecting and organizing information. It may be useful for the teacher to provide scaffolds, such as questions for exploration, and the modelling of graphs and diagrams.
*Living graphs refer to data that is organized by physically moving and arranging students or actual materials in such a way as to show and compare quantities.
NotesAn increasing number of computer and web-based applications are available that enable learners to manipulate data in order to create graphs.
Students should have a lot of experience of organizing data in a variety of ways, and of talking about the advantages and disadvantages of each. Interpretations of data should include the information that cannot be concluded as well as that which can. It is important to remember that the chosen format should illustrate the information without bias.
NotesUsing data that has been collected and saved 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.
Students should have the opportunity to use databases, ideally, those created using data collected by the students then entered into a database by the teacher or together.
NotesA 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. Using a common set of data is a good way for students to start to set up their own databases. A unit of inquiry would be an excellent source of common data for student practice.
Learning continuums
Mathematics scope and sequence 9
Very
you
ng c
hild
ren
view
the
wor
ld a
s a
plac
e of
pos
sibi
litie
s. T
he te
ache
r sho
uld
try
to in
trod
uce
prac
tical
exa
mpl
es a
nd
shou
ld u
se a
ppro
pria
te v
ocab
ular
y.
Dis
cuss
ions
abo
ut c
hanc
e in
dai
ly e
vent
s sh
ould
be
rele
vant
to th
e co
ntex
t of t
he
lear
ners
.
Situ
atio
ns th
at c
ome
up n
atur
ally
in
the
clas
sroo
m, o
ften
thro
ugh
liter
atur
e,
pres
ent o
ppor
tuni
ties
for d
iscu
ssin
g pr
obab
ility
. Dis
cuss
ions
nee
d to
take
pl
ace
in w
hich
stu
dent
s ca
n sh
are
thei
r se
nse
of li
kelih
ood
in te
rms
that
are
use
ful
to th
em.
Situ
atio
ns th
at c
ome
up n
atur
ally
in th
e cl
assr
oom
or f
orm
par
t of t
he u
nits
of
inqu
iry
pres
ent o
ppor
tuni
ties
for s
tude
nts
to fu
rthe
r dev
elop
thei
r und
erst
andi
ng o
f st
atis
tics
and
prob
abili
ty c
once
pts.
Tech
nolo
gy g
ives
us
the
optio
n of
cr
eatin
g a
grap
h at
the
pres
s of
a k
ey.
Bein
g ab
le to
gen
erat
e di
ffer
ent t
ypes
of
gra
phs
allo
ws
lear
ners
to e
xplo
re a
nd
appr
ecia
te th
e at
trib
utes
of e
ach
type
of
grap
h an
d its
eff
icac
y in
dis
play
ing
the
data
.
Tech
nolo
gy a
lso
give
s us
the
poss
ibili
ty
of ra
pidl
y re
plic
atin
g ra
ndom
eve
nts.
Co
mpu
ter a
nd w
eb-b
ased
app
licat
ions
ca
n be
use
d to
toss
coi
ns, r
oll d
ice,
and
ta
bula
te a
nd g
raph
the
resu
lts.
Learning continuums
Mathematics scope and sequence10
MeasurementTo measure is to attach a number to a quantity using a chosen unit. Since the attributes being measured are continuous, 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 expectationsPhase 1Learners will develop an understanding of how measurement involves the comparison of objects and the ordering 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 2Learners will understand that standard units allow us to have a common language to measure and describe 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 more accuracy. Learners will develop these understandings in relation to measurement involving length, mass, capacity, money, temperature and time.
Phase 3Learners will continue to use standard units to measure objects, in particular developing their understanding of measuring perimeter, area and volume. They will select and use appropriate tools and units of measurement, 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 4Learners will understand that a range of procedures exists to measure different attributes of objects and events, for example, the use of formulas for finding area, perimeter and volume. They will be able to decide on 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, the learners will be able to measure and construct angles.
Learning continuums
Mathematics scope and sequence 11
Lear
ning
con
tinu
um fo
r mea
sure
men
t
Phas
e 1
Phas
e 2
Phas
e 3
Phas
e 4
Conc
eptu
al u
nder
stan
din
gsM
easu
rem
ent i
nvol
ves
com
parin
g ob
ject
s an
d ev
ents
.
Obj
ects
hav
e at
trib
utes
that
can
be
mea
sure
d us
ing
non-
stan
dard
uni
ts.
Even
ts c
an b
e or
dere
d an
d se
quen
ced.
Conc
eptu
al u
nder
stan
din
gsSt
anda
rd u
nits
allo
w u
s to
hav
e a
com
mon
la
ngua
ge to
iden
tify,
com
pare
, ord
er a
nd
sequ
ence
obj
ects
and
eve
nts.
We
use
tool
s to
mea
sure
the
attr
ibut
es o
f ob
ject
s an
d ev
ents
.
Estim
atio
n al
low
s us
to m
easu
re w
ith
diff
eren
t lev
els
of a
ccur
acy.
Conc
eptu
al u
nder
stan
din
gsO
bjec
ts a
nd e
vent
s ha
ve a
ttrib
utes
that
ca
n be
mea
sure
d us
ing
appr
opria
te to
ols.
Rela
tions
hips
exi
st b
etw
een
stan
dard
un
its th
at m
easu
re th
e sa
me
attr
ibut
es.
Conc
eptu
al u
nder
stan
din
gsA
ccur
acy
of m
easu
rem
ents
dep
ends
on
the
situ
atio
n an
d th
e pr
ecis
ion
of th
e to
ol.
Conv
ersi
on o
f uni
ts a
nd m
easu
rem
ents
al
low
s us
to m
ake
sens
e of
the
wor
ld w
e liv
e in
.
A ra
nge
of p
roce
dure
s ex
ists
to m
easu
re
diff
eren
t att
ribut
es o
f obj
ects
and
eve
nts.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
at a
ttrib
utes
of r
eal
obje
cts
can
be c
ompa
red
and
desc
ribed
, for
exa
mpl
e, lo
nger
, sh
orte
r, he
avie
r, em
pty,
full,
hot
ter,
cold
er
unde
rsta
nd th
at e
vent
s in
dai
ly
rout
ines
can
be
desc
ribed
and
se
quen
ced,
for e
xam
ple,
bef
ore,
aft
er,
bedt
ime,
sto
rytim
e, to
day,
tom
orro
w.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
e us
e of
sta
ndar
d un
its
to m
easu
re, f
or e
xam
ple,
leng
th, m
ass,
m
oney
, tim
e, te
mpe
ratu
re
unde
rsta
nd th
at to
ols
can
be u
sed
to
mea
sure
unde
rsta
nd th
at c
alen
dars
can
be
used
to d
eter
min
e th
e da
te, a
nd to
id
entif
y an
d se
quen
ce d
ays
of th
e w
eek
and
mon
ths
of th
e ye
ar
unde
rsta
nd th
at ti
me
is m
easu
red
usin
g un
iver
sal u
nits
of m
easu
re, f
or
exam
ple,
yea
rs, m
onth
s, d
ays,
hou
rs,
min
utes
and
sec
onds
.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
e us
e of
sta
ndar
d un
its to
mea
sure
per
imet
er, a
rea
and
volu
me
unde
rsta
nd th
at m
easu
res
can
fall
betw
een
num
bers
on
a m
easu
rem
ent
scal
e, fo
r exa
mpl
e, 3
½ k
g, b
etw
een
4 cm
and
5 c
m
unde
rsta
nd re
latio
nshi
ps b
etw
een
units
, for
exa
mpl
e, m
etre
s,
cent
imet
res
and
mill
imet
res
unde
rsta
nd a
n an
gle
as a
mea
sure
of
rota
tion.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd p
roce
dure
s fo
r fin
ding
ar
ea, p
erim
eter
and
vol
ume
unde
rsta
nd th
e re
latio
nshi
ps b
etw
een
area
and
per
imet
er, b
etw
een
area
and
vo
lum
e, a
nd b
etw
een
volu
me
and
capa
city
unde
rsta
nd u
nit c
onve
rsio
ns w
ithin
m
easu
rem
ent s
yste
ms
(met
ric o
r cu
stom
ary)
.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
iden
tify,
com
pare
and
des
crib
e at
trib
utes
of r
eal o
bjec
ts, f
or e
xam
ple,
lo
nger
, sho
rter
, hea
vier
, em
pty,
full,
ho
tter
, col
der
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
estim
ate
and
mea
sure
obj
ects
usi
ng
stan
dard
uni
ts o
f mea
sure
men
t: le
ngth
, mas
s, c
apac
ity,
mon
ey a
nd
tem
pera
ture
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
estim
ate
and
mea
sure
usi
ng s
tand
ard
units
of m
easu
rem
ent:
perim
eter
, are
a an
d vo
lum
e
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
deve
lop
and
desc
ribe
form
ulas
for
findi
ng p
erim
eter
, are
a an
d vo
lum
e
use
deci
mal
and
frac
tion
nota
tion
in
mea
sure
men
t, fo
r exa
mpl
e, 3
.2 c
m,
1.47
kg,
1½
mile
s
Learning continuums
Mathematics scope and sequence12
com
pare
the
leng
th, m
ass
and
capa
city
of o
bjec
ts u
sing
non
-st
anda
rd u
nits
iden
tify,
des
crib
e an
d se
quen
ce
even
ts in
thei
r dai
ly ro
utin
e, fo
r ex
ampl
e, b
efor
e, a
fter
, bed
time,
st
oryt
ime,
toda
y, to
mor
row
.
read
and
writ
e th
e tim
e to
the
hour
, ha
lf ho
ur a
nd q
uart
er h
our
estim
ate
and
com
pare
leng
ths
of ti
me:
se
cond
, min
ute,
hou
r, da
y, w
eek
and
mon
th.
desc
ribe
mea
sure
s th
at fa
ll be
twee
n nu
mbe
rs o
n a
scal
e
read
and
writ
e di
gita
l and
ana
logu
e tim
e on
12-
hour
and
24-
hour
clo
cks.
read
and
inte
rpre
t sca
les
on a
rang
e of
m
easu
ring
inst
rum
ents
mea
sure
and
con
stru
ct a
ngle
s in
de
gree
s us
ing
a pr
otra
ctor
carr
y ou
t sim
ple
unit
conv
ersi
ons
with
in a
sys
tem
of m
easu
rem
ent
(met
ric o
r cus
tom
ary)
.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
desc
ribe
obse
rvat
ions
abo
ut e
vent
s an
d ob
ject
s in
real
-life
situ
atio
ns
use
non-
stan
dard
uni
ts o
f m
easu
rem
ent t
o so
lve
prob
lem
s in
re
al-li
fe s
ituat
ions
invo
lvin
g le
ngth
, m
ass
and
capa
city
.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
use
stan
dard
uni
ts o
f mea
sure
men
t to
solv
e pr
oble
ms
in re
al-li
fe s
ituat
ions
in
volv
ing
leng
th, m
ass,
cap
acit
y,
mon
ey a
nd te
mpe
ratu
re
use
mea
sure
s of
tim
e to
ass
ist w
ith
prob
lem
sol
ving
in re
al-li
fe s
ituat
ions
.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
use
stan
dard
uni
ts o
f mea
sure
men
t to
solv
e pr
oble
ms
in re
al-li
fe s
ituat
ions
in
volv
ing
perim
eter
, are
a an
d vo
lum
e
sele
ct a
ppro
pria
te to
ols
and
units
of
mea
sure
men
t
use
timel
ines
in u
nits
of i
nqui
ry a
nd
othe
r rea
l-life
situ
atio
ns.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
sele
ct a
nd u
se a
ppro
pria
te u
nits
of
mea
sure
men
t and
tool
s to
sol
ve
prob
lem
s in
real
-life
situ
atio
ns
dete
rmin
e an
d ju
stify
the
leve
l of
accu
racy
requ
ired
to s
olve
real
-life
pr
oble
ms
invo
lvin
g m
easu
rem
ent
use
deci
mal
and
frac
tiona
l not
atio
n in
mea
sure
men
t, fo
r exa
mpl
e, 3
.2 c
m,
1.47
kg,
1½
mile
s
use
timet
able
s an
d sc
hedu
les
(12-
hour
and
24-
hour
clo
cks)
in re
al-li
fe
situ
atio
ns
dete
rmin
e tim
es w
orld
wid
e.
Learning continuums
Mathematics scope and sequence 13
Not
esLe
arne
rs n
eed
man
y op
port
uniti
es to
ex
perie
nce
and
quan
tify
mea
sure
men
t in
a d
irect
kin
esth
etic
man
ner.
They
will
de
velo
p un
ders
tand
ing
of m
easu
rem
ent
by u
sing
man
ipul
ativ
es a
nd m
ater
ials
fr
om th
eir i
mm
edia
te e
nviro
nmen
t, fo
r ex
ampl
e, c
onta
iner
s of
diff
eren
t siz
es,
sand
, wat
er, b
eads
, cor
ks a
nd b
eans
.
Not
esU
sing
mat
eria
ls fr
om th
eir i
mm
edia
te
envi
ronm
ent,
lear
ners
can
inve
stig
ate
how
uni
ts a
re u
sed
for m
easu
rem
ent
and
how
mea
sure
men
ts v
ary
depe
ndin
g on
the
unit
that
is u
sed.
Lea
rner
s w
ill
refin
e th
eir e
stim
atio
n an
d m
easu
rem
ent
skill
s by
bas
ing
estim
atio
ns o
n pr
ior
know
ledg
e, m
easu
ring
the
obje
ct a
nd
com
parin
g ac
tual
mea
sure
men
ts w
ith
thei
r est
imat
ions
.
Not
esIn
ord
er to
use
mea
sure
men
t mor
e au
then
tical
ly, l
earn
ers
shou
ld h
ave
the
oppo
rtun
ity
to m
easu
re re
al o
bjec
ts in
re
al s
ituat
ions
. The
uni
ts o
f inq
uiry
can
of
ten
prov
ide
thes
e re
alis
tic c
onte
xts.
A w
ide
rang
e of
mea
surin
g to
ols
shou
ld
be a
vaila
ble
to th
e st
uden
ts, f
or e
xam
ple,
ru
lers
, tru
ndle
whe
els,
tape
mea
sure
s,
bath
room
sca
les,
kitc
hen
scal
es,
timer
s, a
nalo
gue
cloc
ks, d
igita
l clo
cks,
st
opw
atch
es a
nd c
alen
dars
. The
re a
re a
n in
crea
sing
num
ber o
f com
pute
r and
web
-ba
sed
appl
icat
ions
ava
ilabl
e fo
r stu
dent
s to
use
in a
uthe
ntic
con
text
s.
Plea
se n
ote
that
out
com
es re
latin
g to
an
gles
als
o ap
pear
in th
e sh
ape
and
spac
e st
rand
.
Not
esLe
arne
rs g
ener
aliz
e th
eir m
easu
ring
expe
rienc
es a
s th
ey d
evis
e pr
oced
ures
an
d fo
rmul
as fo
r wor
king
out
per
imet
er,
area
and
vol
ume.
Whi
le th
e em
phas
is fo
r und
erst
andi
ng is
on
mea
sure
men
t sys
tem
s co
mm
only
use
d in
the
lear
ner’s
wor
ld, i
t is
wor
thw
hile
be
ing
awar
e of
the
exis
tenc
e of
oth
er
syst
ems
and
how
con
vers
ions
bet
wee
n sy
stem
s he
lp u
s to
mak
e se
nse
of th
em.
Learning continuums
Mathematics scope and sequence14
Shape and spaceThe regions, paths and boundaries of natural space can be described by shape. An understanding of the interrelationships of shape allows us to interpret, understand and appreciate our two-dimensional (2D) and three-dimensional (3D) world.
Overall expectationsPhase 1Learners 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 2Learners will continue to work with 2D and 3D shapes, developing the understanding that shapes are classified and named according to their properties. They will understand that examples of symmetry and transformations 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 3Learners 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 and describing objects and events in real-world situations.
Phase 4Learners will understand the properties of regular and irregular polyhedra. They will understand the properties 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 will develop their understanding of the use of scale (ratio) to enlarge and reduce shapes. They will apply the language and notation of bearing to describe direction and position.
Learning continuums
Mathematics scope and sequence 15
Lear
ning
con
tinu
um fo
r sha
pe a
nd s
pace
Phas
e 1
Phas
e 2
Phas
e 3
Phas
e 4
Conc
eptu
al u
nder
stan
din
gsSh
apes
can
be
desc
ribed
and
org
aniz
ed
acco
rdin
g to
thei
r pro
pert
ies.
Obj
ects
in o
ur im
med
iate
env
ironm
ent
have
a p
ositi
on in
spa
ce th
at c
an b
e de
scrib
ed a
ccor
ding
to a
poi
nt o
f re
fere
nce.
Conc
eptu
al u
nder
stan
din
gsSh
apes
are
cla
ssifi
ed a
nd n
amed
ac
cord
ing
to th
eir p
rope
rtie
s.
Som
e sh
apes
are
mad
e up
of p
arts
that
re
peat
in s
ome
way
.
Spec
ific
voca
bula
ry c
an b
e us
ed to
de
scrib
e an
obj
ect’s
pos
ition
in s
pace
.
Conc
eptu
al u
nder
stan
din
gsCh
angi
ng th
e po
sitio
n of
a s
hape
doe
s no
t alte
r its
pro
pert
ies.
Shap
es c
an b
e tr
ansf
orm
ed in
diff
eren
t w
ays.
Geo
met
ric s
hape
s an
d vo
cabu
lary
are
us
eful
for r
epre
sent
ing
and
desc
ribin
g ob
ject
s an
d ev
ents
in re
al-w
orld
si
tuat
ions
.
Conc
eptu
al u
nder
stan
din
gsM
anip
ulat
ion
of s
hape
and
spa
ce ta
kes
plac
e fo
r a p
artic
ular
pur
pose
.
Cons
olid
atin
g w
hat w
e kn
ow o
f ge
omet
ric c
once
pts
allo
ws
us to
mak
e se
nse
of a
nd in
tera
ct w
ith o
ur w
orld
.
Geo
met
ric to
ols
and
met
hods
can
be
used
to s
olve
pro
blem
s re
latin
g to
sha
pe
and
spac
e.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
at 2
D a
nd 3
D s
hape
s ha
ve c
hara
cter
istic
s th
at c
an b
e de
scrib
ed a
nd c
ompa
red
unde
rsta
nd th
at c
omm
on la
ngua
ge
can
be u
sed
to d
escr
ibe
posi
tion
and
dire
ctio
n, fo
r exa
mpl
e, in
side
, out
side
, ab
ove,
bel
ow, n
ext t
o, b
ehin
d, in
fron
t of
, up,
dow
n.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
at th
ere
are
rela
tions
hips
am
ong
and
betw
een
2D
and
3D s
hape
s
unde
rsta
nd th
at 2
D a
nd 3
D s
hape
s ca
n be
cre
ated
by
putt
ing
toge
ther
an
d/or
taki
ng a
part
oth
er s
hape
s
unde
rsta
nd th
at e
xam
ples
of
sym
met
ry a
nd tr
ansf
orm
atio
ns c
an b
e fo
und
in th
eir i
mm
edia
te e
nviro
nmen
t
unde
rsta
nd th
at g
eom
etric
sha
pes
are
usef
ul fo
r rep
rese
ntin
g re
al-w
orld
si
tuat
ions
unde
rsta
nd th
at d
irect
ions
can
be
used
to d
escr
ibe
path
way
s, re
gion
s,
posi
tions
and
bou
ndar
ies
of th
eir
imm
edia
te e
nviro
nmen
t.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
e co
mm
on la
ngua
ge
used
to d
escr
ibe
shap
es
unde
rsta
nd th
e pr
oper
ties
of re
gula
r an
d irr
egul
ar p
olyg
ons
unde
rsta
nd c
ongr
uent
or s
imila
r sh
apes
unde
rsta
nd th
at li
nes
and
axes
of
refle
ctiv
e an
d ro
tatio
nal s
ymm
etry
as
sist
with
the
cons
truc
tion
of s
hape
s
unde
rsta
nd a
n an
gle
as a
mea
sure
of
rota
tion
unde
rsta
nd th
at d
irect
ions
for
loca
tion
can
be re
pres
ente
d by
co
ordi
nate
s on
a g
rid
unde
rsta
nd th
at v
isua
lizat
ion
of s
hape
an
d sp
ace
is a
str
ateg
y fo
r sol
ving
pr
oble
ms.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
e co
mm
on la
ngua
ge
used
to d
escr
ibe
shap
es
unde
rsta
nd th
e pr
oper
ties
of re
gula
r an
d irr
egul
ar p
olyh
edra
unde
rsta
nd th
e pr
oper
ties
of c
ircle
s
unde
rsta
nd h
ow s
cale
(rat
ios)
is u
sed
to e
nlar
ge a
nd re
duce
sha
pes
unde
rsta
nd s
yste
ms
for d
escr
ibin
g po
sitio
n an
d di
rect
ion
unde
rsta
nd th
at 2
D re
pres
enta
tions
of
3D
obj
ects
can
be
used
to v
isua
lize
and
solv
e pr
oble
ms
unde
rsta
nd th
at g
eom
etric
idea
s an
d re
latio
nshi
ps c
an b
e us
ed to
so
lve
prob
lem
s in
oth
er a
reas
of
mat
hem
atic
s an
d in
real
life
.
Learning continuums
Mathematics scope and sequence16
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
sort
, des
crib
e an
d co
mpa
re 3
D s
hape
s
desc
ribe
posi
tion
and
dire
ctio
n, fo
r ex
ampl
e, in
side
, out
side
, abo
ve,
belo
w, n
ext t
o, b
ehin
d, in
fron
t of,
up,
dow
n.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
sort
, des
crib
e an
d la
bel 2
D a
nd 3
D
shap
es
anal
yse
and
desc
ribe
the
rela
tions
hips
be
twee
n 2D
and
3D
sha
pes
crea
te a
nd d
escr
ibe
sym
met
rical
and
te
ssel
latin
g pa
tter
ns
iden
tify
lines
of r
efle
ctiv
e sy
mm
etry
repr
esen
t ide
as a
bout
the
real
wor
ld
usin
g ge
omet
ric v
ocab
ular
y an
d sy
mbo
ls, f
or e
xam
ple,
thro
ugh
oral
de
scrip
tion,
dra
win
g, m
odel
ling,
la
belli
ng
inte
rpre
t and
cre
ate
sim
ple
dire
ctio
ns,
desc
ribin
g pa
ths,
regi
ons,
pos
ition
s an
d bo
unda
ries
of th
eir i
mm
edia
te
envi
ronm
ent.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
sort
, des
crib
e an
d m
odel
regu
lar a
nd
irreg
ular
pol
ygon
s
desc
ribe
and
mod
el c
ongr
uenc
y an
d si
mila
rity
in 2
D s
hape
s
anal
yse
angl
es b
y co
mpa
ring
and
desc
ribin
g ro
tatio
ns: w
hole
turn
; hal
f tu
rn; q
uart
er tu
rn; n
orth
, sou
th, e
ast
and
wes
t on
a co
mpa
ss
loca
te fe
atur
es o
n a
grid
usi
ng
coor
dina
tes
desc
ribe
and/
or re
pres
ent m
enta
l im
ages
of o
bjec
ts, p
atte
rns,
and
pa
ths.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
anal
yse,
des
crib
e, c
lass
ify a
nd v
isua
lize
2D (i
nclu
ding
circ
les,
tria
ngle
s an
d qu
adril
ater
als)
and
3D
sha
pes,
usi
ng
geom
etric
voc
abul
ary
desc
ribe
lines
and
ang
les
usin
g ge
omet
ric v
ocab
ular
y
iden
tify
and
use
scal
e (ra
tios)
to
enla
rge
and
redu
ce s
hape
s
iden
tify
and
use
the
lang
uage
and
no
tatio
n of
bea
ring
to d
escr
ibe
dire
ctio
n an
d po
sitio
n
crea
te a
nd m
odel
how
a 2
D n
et
conv
erts
into
a 3
D s
hape
and
vic
e ve
rsa
expl
ore
the
use
of g
eom
etric
idea
s an
d re
latio
nshi
ps to
sol
ve p
robl
ems
in
othe
r are
as o
f mat
hem
atic
s.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
expl
ore
and
desc
ribe
the
path
s,
regi
ons
and
boun
darie
s of
thei
r im
med
iate
env
ironm
ent (
insi
de,
outs
ide,
abo
ve, b
elow
) and
thei
r po
sitio
n (n
ext t
o, b
ehin
d, in
fron
t of,
up, d
own)
.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
anal
yse
and
use
wha
t the
y kn
ow
abou
t 3D
sha
pes
to d
escr
ibe
and
wor
k w
ith 2
D s
hape
s
reco
gniz
e an
d ex
plai
n si
mpl
e sy
mm
etric
al d
esig
ns in
the
envi
ronm
ent
appl
y kn
owle
dge
of s
ymm
etry
to
prob
lem
-sol
ving
situ
atio
ns
inte
rpre
t and
use
sim
ple
dire
ctio
ns,
desc
ribin
g pa
ths,
regi
ons,
pos
ition
s an
d bo
unda
ries
of th
eir i
mm
edia
te
envi
ronm
ent.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
anal
yse
and
desc
ribe
2D a
nd 3
D
shap
es, i
nclu
ding
regu
lar a
nd
irreg
ular
pol
ygon
s, u
sing
geo
met
rical
vo
cabu
lary
iden
tify,
des
crib
e an
d m
odel
co
ngru
ency
and
sim
ilarit
y in
2D
sh
apes
reco
gniz
e an
d ex
plai
n sy
mm
etric
al
patt
erns
, inc
ludi
ng te
ssel
latio
n, in
the
envi
ronm
ent
appl
y kn
owle
dge
of tr
ansf
orm
atio
ns
to p
robl
em-s
olvi
ng s
ituat
ions
.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
use
geom
etric
voc
abul
ary
whe
n de
scrib
ing
shap
e an
d sp
ace
in
mat
hem
atic
al s
ituat
ions
and
bey
ond
use
scal
e (ra
tios)
to e
nlar
ge a
nd
redu
ce s
hape
s
appl
y th
e la
ngua
ge a
nd n
otat
ion
of
bear
ing
to d
escr
ibe
dire
ctio
n an
d po
sitio
n
use
2D re
pres
enta
tions
of 3
D o
bjec
ts
to v
isua
lize
and
solv
e pr
oble
ms,
for
exam
ple
usin
g dr
awin
gs o
r mod
els.
Learning continuums
Mathematics scope and sequence 17
Not
esLe
arne
rs n
eed
man
y op
port
uniti
es to
ex
perie
nce
shap
e an
d sp
ace
in a
dire
ct
kine
sthe
tic m
anne
r, fo
r exa
mpl
e, th
roug
h pl
ay, c
onst
ruct
ion
and
mov
emen
t.
The
man
ipul
ativ
es th
at th
ey in
tera
ct w
ith
shou
ld in
clud
e a
rang
e of
3D
sha
pes,
in
part
icul
ar th
e re
al-li
fe o
bjec
ts w
ith w
hich
ch
ildre
n ar
e fa
mili
ar. 2
D s
hape
s (p
lane
sh
apes
) are
a m
ore
abst
ract
con
cept
but
ca
n be
und
erst
ood
as fa
ces
of 3
D s
hape
s.
Not
esLe
arne
rs n
eed
to u
nder
stan
d th
e pr
oper
ties
of 2
D a
nd 3
D s
hape
s be
fore
the
mat
hem
atic
al v
ocab
ular
y as
soci
ated
with
sha
pes
mak
es s
ense
to
them
. Thr
ough
cre
atin
g an
d m
anip
ulat
ing
shap
es, l
earn
ers
alig
n th
eir
natu
ral v
ocab
ular
y w
ith m
ore
form
al
mat
hem
atic
al v
ocab
ular
y an
d be
gin
to
appr
ecia
te th
e ne
ed fo
r thi
s pr
ecis
ion.
Not
esCo
mpu
ter a
nd w
eb-b
ased
app
licat
ions
ca
n be
use
d to
exp
lore
sha
pe a
nd s
pace
co
ncep
ts s
uch
as s
ymm
etry
, ang
les
and
coor
dina
tes.
The
units
of i
nqui
ry c
an p
rovi
de a
uthe
ntic
co
ntex
ts fo
r dev
elop
ing
unde
rsta
ndin
g of
con
cept
s re
latin
g to
loca
tion
and
dire
ctio
ns.
Not
esTo
ols
such
as
com
pass
es a
nd p
rotr
acto
rs
are
com
mon
ly u
sed
to s
olve
pro
blem
s in
re
al-li
fe s
ituat
ions
. How
ever
, car
e sh
ould
be
take
n to
ens
ure
that
stu
dent
s ha
ve a
st
rong
und
erst
andi
ng o
f the
con
cept
s em
bedd
ed in
the
prob
lem
to e
nsur
e m
eani
ngfu
l eng
agem
ent w
ith th
e to
ols
and
full
unde
rsta
ndin
g of
the
solu
tion.
Learning continuums
Mathematics scope and sequence18
Pattern and functionTo 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”. This builds a foundation for the later study of algebra.
Overall expectations Phase 1Learners will understand that patterns and sequences occur in everyday situations. They will be able to identify, describe, extend and create patterns in various ways.
Phase 2Learners 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 and commutative 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 and subtraction.
Phase 3Learners 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 another set. They will understand the inverse relationship between multiplication and division, and the associative and commutative properties of multiplication. They will be able to use their understanding of pattern and function to represent and make sense of real-life situations and, where appropriate, to solve problems involving the four operations.
Phase 4Learners 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 to express 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 of real-life situations and to solve problems involving the four operations.
Learning continuums
Mathematics scope and sequence 19
Lear
ning
con
tinu
um fo
r pat
tern
and
func
tion
Phas
e 1
Phas
e 2
Phas
e 3
Phas
e 4
Conc
eptu
al u
nder
stan
din
gsPa
tter
ns a
nd s
eque
nces
occ
ur in
eve
ryda
y si
tuat
ions
.
Patt
erns
repe
at a
nd g
row
.
Conc
eptu
al u
nder
stan
din
gsW
hole
num
bers
exh
ibit
patt
erns
and
re
latio
nshi
ps th
at c
an b
e ob
serv
ed a
nd
desc
ribed
.
Patt
erns
can
be
repr
esen
ted
usin
g nu
mbe
rs a
nd o
ther
sym
bols
.
Conc
eptu
al u
nder
stan
din
gsFu
nctio
ns a
re re
latio
nshi
ps o
r rul
es th
at
uniq
uely
ass
ocia
te m
embe
rs o
f one
set
w
ith m
embe
rs o
f ano
ther
set
.
By a
naly
sing
pat
tern
s an
d id
entif
ying
ru
les
for p
atte
rns
it is
pos
sibl
e to
mak
e pr
edic
tions
.
Conc
eptu
al u
nder
stan
din
gsPa
tter
ns c
an o
ften
be
gene
raliz
ed u
sing
al
gebr
aic
expr
essi
ons,
equ
atio
ns o
r fu
nctio
ns.
Expo
nent
ial n
otat
ion
is a
pow
erfu
l way
to
exp
ress
repe
ated
pro
duct
s of
the
sam
e nu
mbe
r.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
at p
atte
rns
can
be
foun
d in
eve
ryda
y si
tuat
ions
, for
ex
ampl
e, s
ound
s, a
ctio
ns, o
bjec
ts,
natu
re.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
at p
atte
rns
can
be
foun
d in
num
bers
, for
exa
mpl
e, o
dd
and
even
num
bers
, ski
p co
untin
g
unde
rsta
nd th
e in
vers
e re
latio
nshi
p be
twee
n ad
ditio
n an
d su
btra
ctio
n
unde
rsta
nd th
e as
soci
ativ
e an
d co
mm
utat
ive
prop
ertie
s of
add
ition
.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
at p
atte
rns
can
be
anal
ysed
and
rule
s id
entif
ied
unde
rsta
nd th
at m
ultip
licat
ion
is
repe
ated
add
ition
and
that
div
isio
n is
re
peat
ed s
ubtr
actio
n
unde
rsta
nd th
e in
vers
e re
latio
nshi
p be
twee
n m
ultip
licat
ion
and
divi
sion
unde
rsta
nd th
e as
soci
ativ
e an
d co
mm
utat
ive
prop
ertie
s of
m
ultip
licat
ion.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd th
at p
atte
rns
can
be
gene
raliz
ed b
y a
rule
unde
rsta
nd e
xpon
ents
as
repe
ated
m
ultip
licat
ion
unde
rsta
nd th
e in
vers
e re
latio
nshi
p be
twee
n ex
pone
nts
and
root
s
unde
rsta
nd th
at p
atte
rns
can
be
repr
esen
ted,
ana
lyse
d an
d ge
nera
lized
us
ing
tabl
es, g
raph
s, w
ords
, and
, w
hen
poss
ible
, sym
bolic
rule
s.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
desc
ribe
patt
erns
in v
ario
us w
ays,
fo
r exa
mpl
e, u
sing
wor
ds, d
raw
ings
, sy
mbo
ls, m
ater
ials
, act
ions
, num
bers
.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
repr
esen
t pat
tern
s in
a v
arie
ty o
f way
s,
for e
xam
ple,
usi
ng w
ords
, dra
win
gs,
sym
bols
, mat
eria
ls, a
ctio
ns, n
umbe
rs
desc
ribe
num
ber p
atte
rns,
for
exam
ple,
odd
and
eve
n nu
mbe
rs, s
kip
coun
ting.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
desc
ribe
the
rule
for a
pat
tern
in a
va
riety
of w
ays
repr
esen
t rul
es fo
r pat
tern
s us
ing
wor
ds, s
ymbo
ls a
nd ta
bles
iden
tify
a se
quen
ce o
f ope
ratio
ns
rela
ting
one
set o
f num
bers
to a
noth
er
set.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
repr
esen
t the
rule
of a
pat
tern
by
usin
g a
func
tion
anal
yse
patt
ern
and
func
tion
usin
g w
ords
, tab
les
and
grap
hs, a
nd, w
hen
poss
ible
, sym
bolic
rule
s.
Learning continuums
Mathematics scope and sequence20
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
exte
nd a
nd c
reat
e pa
tter
ns.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
exte
nd a
nd c
reat
e pa
tter
ns in
nu
mbe
rs, f
or e
xam
ple,
odd
and
eve
n nu
mbe
rs, s
kip
coun
ting
use
num
ber p
atte
rns
to re
pres
ent a
nd
unde
rsta
nd re
al-li
fe s
ituat
ions
use
the
prop
ertie
s an
d re
latio
nshi
ps
of a
dditi
on a
nd s
ubtr
actio
n to
sol
ve
prob
lem
s.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
sele
ct a
ppro
pria
te m
etho
ds fo
r re
pres
entin
g pa
tter
ns, f
or e
xam
ple
usin
g w
ords
, sym
bols
and
tabl
es
use
num
ber p
atte
rns
to m
ake
pred
ictio
ns a
nd s
olve
pro
blem
s
use
the
prop
ertie
s an
d re
latio
nshi
ps o
f th
e fo
ur o
pera
tions
to s
olve
pro
blem
s.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
sele
ct a
ppro
pria
te m
etho
ds to
ana
lyse
pa
tter
ns a
nd id
entif
y ru
les
use
func
tions
to s
olve
pro
blem
s.
Not
esTh
e w
orld
is fi
lled
with
pat
tern
and
ther
e w
ill b
e m
any
oppo
rtun
ities
for l
earn
ers
to m
ake
this
con
nect
ion
acro
ss th
e cu
rric
ulum
.
A ra
nge
of m
anip
ulat
ives
can
be
used
to
expl
ore
patt
erns
incl
udin
g pa
tter
n bl
ocks
, at
trib
ute
bloc
ks, c
olou
r tile
s, c
alcu
lato
rs,
num
ber c
hart
s, b
eans
and
but
tons
.
Not
esSt
uden
ts w
ill a
pply
thei
r und
erst
andi
ng
of p
atte
rn to
the
num
bers
they
alre
ady
know
. The
pat
tern
s th
ey fi
nd w
ill h
elp
to
deep
en th
eir u
nder
stan
ding
of a
rang
e of
nu
mbe
r con
cept
s.
Four
-fun
ctio
n ca
lcul
ator
s ca
n be
use
d to
ex
plor
e nu
mbe
r pat
tern
s.
Not
esPa
tter
ns a
re c
entr
al to
the
unde
rsta
ndin
g of
all
conc
epts
in m
athe
mat
ics.
The
y ar
e th
e ba
sis
of h
ow o
ur n
umbe
r sys
tem
is
orga
nize
d. S
earc
hing
for,
and
iden
tifyi
ng,
patt
erns
hel
ps u
s to
see
rela
tions
hips
, m
ake
gene
raliz
atio
ns, a
nd is
a p
ower
ful
stra
tegy
for p
robl
em s
olvi
ng. F
unct
ions
de
velo
p fr
om th
e st
udy
of p
atte
rns
and
mak
e it
poss
ible
to p
redi
ct in
mat
hem
atic
s pr
oble
ms.
Not
esA
lgeb
ra is
a m
athe
mat
ical
lang
uage
us
ing
num
bers
and
sym
bols
to e
xpre
ss
rela
tions
hips
. Whe
n th
e sa
me
rela
tions
hip
wor
ks w
ith a
ny n
umbe
r, al
gebr
a us
es
lett
ers
to re
pres
ent t
he g
ener
aliz
atio
n.
Lett
ers
can
be u
sed
to re
pres
ent t
he
quan
tity.
Learning continuums
Mathematics scope and sequence 21
NumberOur 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 operations of 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 the result will be used.
Overall expectationsPhase 1Learners will understand that numbers are used for many different purposes in the real world. They will develop an understanding of one-to-one correspondence and conservation of number, and be able to count and use number words and numerals to represent quantities.
Phase 2Learners 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 addition and 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 understanding of fractions as representations of whole-part relationships and will be able to model fractions and use fraction names in real-life situations.
Phase 3Learners 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, and use them in real-life situations. Learners will have automatic recall of addition, subtraction, multiplication and 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 reasonableness of their answers.
Phase 4Learners 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.
Learning continuums
Mathematics scope and sequence22
Lear
ning
con
tinu
um fo
r num
ber
Phas
e 1
Phas
e 2
Phas
e 3
Phas
e 4
Conc
eptu
al u
nder
stan
din
gsN
umbe
rs a
re a
nam
ing
syst
em.
Num
bers
can
be
used
in m
any
way
s fo
r di
ffer
ent p
urpo
ses
in th
e re
al w
orld
.
Num
bers
are
con
nect
ed to
eac
h ot
her
thro
ugh
a va
riety
of r
elat
ions
hips
.
Mak
ing
conn
ectio
ns b
etw
een
our
expe
rienc
es w
ith n
umbe
r can
hel
p us
to
deve
lop
num
ber s
ense
.
Conc
eptu
al u
nder
stan
din
gsTh
e ba
se 1
0 pl
ace
valu
e sy
stem
is u
sed
to re
pres
ent n
umbe
rs a
nd n
umbe
r re
latio
nshi
ps.
Frac
tions
are
way
s of
repr
esen
ting
who
le-
part
rela
tions
hips
.
The
oper
atio
ns o
f add
ition
, sub
trac
tion,
m
ultip
licat
ion
and
divi
sion
are
rela
ted
to e
ach
othe
r and
are
use
d to
pro
cess
in
form
atio
n to
sol
ve p
robl
ems.
Num
ber o
pera
tions
can
be
mod
elle
d in
a
varie
ty o
f way
s.
Ther
e ar
e m
any
men
tal m
etho
ds th
at c
an
be a
pplie
d fo
r exa
ct a
nd a
ppro
xim
ate
com
puta
tions
.
Conc
eptu
al u
nder
stan
din
gsTh
e ba
se 1
0 pl
ace
valu
e sy
stem
can
be
exte
nded
to re
pres
ent m
agni
tude
.
Frac
tions
and
dec
imal
s ar
e w
ays
of
repr
esen
ting
who
le-p
art r
elat
ions
hips
.
The
oper
atio
ns o
f add
ition
, sub
trac
tion,
m
ultip
licat
ion
and
divi
sion
are
rela
ted
to e
ach
othe
r and
are
use
d to
pro
cess
in
form
atio
n to
sol
ve p
robl
ems.
Even
com
plex
ope
ratio
ns c
an b
e m
odel
led
in a
var
iety
of w
ays,
for
exam
ple,
an
algo
rithm
is a
way
to
repr
esen
t an
oper
atio
n.
Conc
eptu
al u
nder
stan
din
gsTh
e ba
se 1
0 pl
ace
valu
e sy
stem
ext
ends
in
finite
ly in
two
dire
ctio
ns.
Frac
tions
, dec
imal
frac
tions
and
pe
rcen
tage
s ar
e w
ays
of re
pres
entin
g w
hole
-par
t rel
atio
nshi
ps.
For f
ract
iona
l and
dec
imal
com
puta
tion,
th
e id
eas
deve
lope
d fo
r who
le-n
umbe
r co
mpu
tatio
n ca
n ap
ply.
Ratio
s ar
e a
com
paris
on o
f tw
o nu
mbe
rs
or q
uant
ities
.
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
unde
rsta
nd o
ne-t
o-on
e co
rres
pond
ence
unde
rsta
nd th
at, f
or a
set
of o
bjec
ts,
the
num
ber n
ame
of th
e la
st o
bjec
t co
unte
d de
scrib
es th
e qu
antit
y of
the
who
le s
et
unde
rsta
nd th
at n
umbe
rs c
an
be c
onst
ruct
ed in
mul
tiple
way
s,
for e
xam
ple,
by
com
bini
ng a
nd
part
ition
ing
unde
rsta
nd c
onse
rvat
ion
of n
umbe
r*
unde
rsta
nd th
e re
lativ
e m
agni
tude
of
who
le n
umbe
rs
reco
gniz
e gr
oups
of z
ero
to fi
ve
obje
cts
with
out c
ount
ing
(sub
itizi
ng)
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
mod
el n
umbe
rs to
hun
dred
s or
be
yond
usi
ng th
e ba
se 1
0 pl
ace
valu
e sy
stem
**
estim
ate
quan
titie
s to
100
or b
eyon
d
mod
el s
impl
e fr
actio
n re
latio
nshi
ps
use
the
lang
uage
of a
dditi
on a
nd
subt
ract
ion,
for e
xam
ple,
add
, tak
e aw
ay, p
lus,
min
us, s
um, d
iffer
ence
mod
el a
dditi
on a
nd s
ubtr
actio
n of
w
hole
num
bers
deve
lop
stra
tegi
es fo
r mem
oriz
ing
addi
tion
and
subt
ract
ion
num
ber
fact
s
estim
ate
sum
s an
d di
ffer
ence
s
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
mod
el n
umbe
rs to
thou
sand
s or
be
yond
usi
ng th
e ba
se 1
0 pl
ace
valu
e sy
stem
mod
el e
quiv
alen
t fra
ctio
ns
use
the
lang
uage
of f
ract
ions
, for
ex
ampl
e, n
umer
ator
, den
omin
ator
mod
el d
ecim
al fr
actio
ns to
hu
ndre
dths
or b
eyon
d
mod
el m
ultip
licat
ion
and
divi
sion
of
who
le n
umbe
rs
use
the
lang
uage
of m
ultip
licat
ion
and
divi
sion
, for
exa
mpl
e, fa
ctor
, m
ultip
le, p
rodu
ct, q
uotie
nt, p
rime
num
bers
, com
posi
te n
umbe
r
Lear
ning
out
com
esW
hen
cons
truc
ting
mea
ning
lear
ners
:
mod
el n
umbe
rs to
mill
ions
or b
eyon
d us
ing
the
base
10
plac
e va
lue
syst
em
mod
el ra
tios
mod
el in
tege
rs in
app
ropr
iate
co
ntex
ts
mod
el e
xpon
ents
and
squ
are
root
s
mod
el im
prop
er fr
actio
ns a
nd m
ixed
nu
mbe
rs
sim
plify
frac
tions
usi
ng m
anip
ulat
ives
mod
el d
ecim
al fr
actio
ns to
th
ousa
ndth
s or
bey
ond
mod
el p
erce
ntag
es
unde
rsta
nd th
e re
latio
nshi
p be
twee
n fr
actio
ns, d
ecim
als
and
perc
enta
ges
Learning continuums
Mathematics scope and sequence 23
unde
rsta
nd w
hole
-par
t rel
atio
nshi
ps
use
the
lang
uage
of m
athe
mat
ics
to c
ompa
re q
uant
ities
, for
exa
mpl
e,
mor
e, le
ss, f
irst,
seco
nd.
unde
rsta
nd s
ituat
ions
that
invo
lve
mul
tiplic
atio
n an
d di
visi
on
mod
el a
dditi
on a
nd s
ubtr
actio
n of
fr
actio
ns w
ith th
e sa
me
deno
min
ator
.
mod
el a
dditi
on a
nd s
ubtr
actio
n of
frac
tions
with
rela
ted
deno
min
ator
s***
mod
el a
dditi
on a
nd s
ubtr
actio
n of
de
cim
als.
mod
el a
dditi
on, s
ubtr
actio
n,
mul
tiplic
atio
n an
d di
visi
on o
f fra
ctio
ns
mod
el a
dditi
on, s
ubtr
actio
n,
mul
tiplic
atio
n an
d di
visi
on o
f de
cim
als.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
conn
ect n
umbe
r nam
es a
nd
num
eral
s to
the
quan
titie
s th
ey
repr
esen
t.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
read
and
writ
e w
hole
num
bers
up
to
hund
reds
or b
eyon
d
read
, writ
e, c
ompa
re a
nd o
rder
ca
rdin
al a
nd o
rdin
al n
umbe
rs
desc
ribe
men
tal a
nd w
ritte
n st
rate
gies
for a
ddin
g an
d su
btra
ctin
g tw
o-di
git n
umbe
rs.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
read
, writ
e, c
ompa
re a
nd o
rder
who
le
num
bers
up
to th
ousa
nds
or b
eyon
d
deve
lop
stra
tegi
es fo
r mem
oriz
ing
addi
tion,
sub
trac
tion,
mul
tiplic
atio
n an
d di
visi
on n
umbe
r fac
ts
read
, writ
e, c
ompa
re a
nd o
rder
fr
actio
ns
read
and
writ
e eq
uiva
lent
frac
tions
read
, writ
e, c
ompa
re a
nd o
rder
fr
actio
ns to
hun
dred
ths
or b
eyon
d
desc
ribe
men
tal a
nd w
ritte
n st
rate
gies
for m
ultip
licat
ion
and
divi
sion
.
Whe
n tr
ansf
erri
ng m
eani
ng in
to
sym
bol
s le
arne
rs:
read
, writ
e, c
ompa
re a
nd o
rder
who
le
num
bers
up
to m
illio
ns o
r bey
ond
read
and
writ
e ra
tios
read
and
writ
e in
tege
rs in
app
ropr
iate
co
ntex
ts
read
and
writ
e ex
pone
nts
and
squa
re
root
s
conv
ert i
mpr
oper
frac
tions
to m
ixed
nu
mbe
rs a
nd v
ice
vers
a
sim
plify
frac
tions
in m
enta
l and
w
ritte
n fo
rm
read
, writ
e, c
ompa
re a
nd o
rder
de
cim
al fr
actio
ns to
thou
sand
ths
or
beyo
nd
read
, writ
e, c
ompa
re a
nd o
rder
pe
rcen
tage
s
conv
ert b
etw
een
frac
tions
, dec
imal
s an
d pe
rcen
tage
s.
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
coun
t to
dete
rmin
e th
e nu
mbe
r of
obje
cts
in a
set
use
num
ber w
ords
and
num
eral
s to
repr
esen
t qua
ntiti
es in
real
-life
si
tuat
ions
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
use
who
le n
umbe
rs u
p to
hun
dred
s or
be
yond
in re
al-li
fe s
ituat
ions
use
card
inal
and
ord
inal
num
bers
in
real
-life
situ
atio
ns
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
use
who
le n
umbe
rs u
p to
thou
sand
s or
bey
ond
in re
al-li
fe s
ituat
ions
use
fast
reca
ll of
mul
tiplic
atio
n an
d di
visi
on n
umbe
r fac
ts in
real
-life
si
tuat
ions
Whe
n ap
ply
ing
wit
h un
der
stan
din
g le
arne
rs:
use
who
le n
umbe
rs u
p to
mill
ions
or
beyo
nd in
real
-life
situ
atio
ns
use
ratio
s in
real
-life
situ
atio
ns
use
inte
gers
in re
al-li
fe s
ituat
ions
Learning continuums
Mathematics scope and sequence24
use
the
lang
uage
of m
athe
mat
ics
to c
ompa
re q
uant
ities
in re
al-li
fe
situ
atio
ns, f
or e
xam
ple,
mor
e, le
ss,
first
, sec
ond
subi
tize
in re
al-li
fe s
ituat
ions
use
sim
ple
frac
tion
nam
es in
real
-life
si
tuat
ions
.
use
fast
reca
ll of
add
ition
and
su
btra
ctio
n nu
mbe
r fac
ts in
real
-life
si
tuat
ions
use
frac
tions
in re
al-li
fe s
ituat
ions
use
men
tal a
nd w
ritte
n st
rate
gies
for
addi
tion
and
subt
ract
ion
of tw
o-di
git n
umbe
rs o
r bey
ond
in re
al-li
fe
situ
atio
ns
sele
ct a
n ap
prop
riate
met
hod
for
solv
ing
a pr
oble
m, f
or e
xam
ple,
m
enta
l est
imat
ion,
men
tal o
r writ
ten
stra
tegi
es, o
r by
usin
g a
calc
ulat
or
use
stra
tegi
es to
eva
luat
e th
e re
ason
able
ness
of a
nsw
ers.
use
deci
mal
frac
tions
in re
al-li
fe
situ
atio
ns
use
men
tal a
nd w
ritte
n st
rate
gies
for
mul
tiplic
atio
n an
d di
visi
on in
real
-life
si
tuat
ions
sele
ct a
n ef
ficie
nt m
etho
d fo
r so
lvin
g a
prob
lem
, for
exa
mpl
e,
men
tal e
stim
atio
n, m
enta
l or w
ritte
n st
rate
gies
, or b
y us
ing
a ca
lcul
ator
use
stra
tegi
es to
eva
luat
e th
e re
ason
able
ness
of a
nsw
ers
add
and
subt
ract
frac
tions
with
re
late
d de
nom
inat
ors
in re
al-li
fe
situ
atio
ns
add
and
subt
ract
dec
imal
s in
real
-life
si
tuat
ions
, inc
ludi
ng m
oney
estim
ate
sum
, diff
eren
ce, p
rodu
ct
and
quot
ient
in re
al-li
fe s
ituat
ions
, in
clud
ing
frac
tions
and
dec
imal
s.
conv
ert i
mpr
oper
frac
tions
to m
ixed
nu
mbe
rs a
nd v
ice
vers
a in
real
-life
si
tuat
ions
sim
plify
frac
tions
in c
ompu
tatio
n an
swer
s
use
frac
tions
, dec
imal
s an
d pe
rcen
tage
s in
terc
hang
eabl
y in
real
-lif
e si
tuat
ions
sele
ct a
nd u
se a
n ap
prop
riate
se
quen
ce o
f ope
ratio
ns to
sol
ve w
ord
prob
lem
s
sele
ct a
n ef
ficie
nt m
etho
d fo
r sol
ving
a
prob
lem
: men
tal e
stim
atio
n, m
enta
l co
mpu
tatio
n, w
ritte
n al
gorit
hms,
by
usin
g a
calc
ulat
or
use
stra
tegi
es to
eva
luat
e th
e re
ason
able
ness
of a
nsw
ers
use
men
tal a
nd w
ritte
n st
rate
gies
for
addi
ng, s
ubtr
actin
g, m
ultip
lyin
g an
d di
vidi
ng fr
actio
ns a
nd d
ecim
als
in
real
-life
situ
atio
ns
estim
ate
and
mak
e ap
prox
imat
ions
in
real
-life
situ
atio
ns in
volv
ing
frac
tions
, de
cim
als
and
perc
enta
ges.
Learning continuums
Mathematics scope and sequence 25
Not
es*T
o co
nser
ve, i
n m
athe
mat
ical
term
s,
mea
ns th
e am
ount
sta
ys th
e sa
me
rega
rdle
ss o
f the
arr
ange
men
t.
Lear
ners
who
hav
e be
en e
ncou
rage
d to
sel
ect t
heir
own
appa
ratu
s an
d m
etho
ds, a
nd w
ho b
ecom
e ac
cust
omed
to
dis
cuss
ing
and
ques
tioni
ng th
eir
wor
k, w
ill h
ave
conf
iden
ce in
look
ing
for
alte
rnat
ive
appr
oach
es w
hen
an in
itial
at
tem
pt is
uns
ucce
ssfu
l.
Estim
atio
n is
a s
kill
that
will
dev
elop
w
ith e
xper
ienc
e an
d w
ill h
elp
child
ren
gain
a “f
eel”
for n
umbe
rs. C
hild
ren
mus
t be
giv
en th
e op
port
unit
y to
che
ck th
eir
estim
ates
so
that
they
are
abl
e to
furt
her
refin
e an
d im
prov
e th
eir e
stim
atio
n sk
ills.
Ther
e ar
e m
any
oppo
rtun
ities
in th
e un
its
of in
quir
y an
d du
ring
the
scho
ol d
ay fo
r st
uden
ts to
pra
ctis
e an
d ap
ply
num
ber
conc
epts
aut
hent
ical
ly.
Not
es**
Mod
ellin
g in
volv
es u
sing
con
cret
e m
ater
ials
to re
pres
ent n
umbe
rs o
r nu
mbe
r ope
ratio
ns, f
or e
xam
ple,
the
use
of p
atte
rn b
lock
s or
frac
tion
piec
es to
re
pres
ent f
ract
ions
and
the
use
of b
ase
10
bloc
ks to
repr
esen
t num
ber o
pera
tions
.
Stud
ents
nee
d to
use
num
bers
in
man
y si
tuat
ions
in o
rder
to a
pply
thei
r un
ders
tand
ing
to n
ew s
ituat
ions
. In
addi
tion
to th
e un
its o
f inq
uiry
, chi
ldre
n’s
liter
atur
e al
so p
rovi
des
rich
oppo
rtun
ities
fo
r dev
elop
ing
num
ber c
once
pts.
To b
e us
eful
, add
ition
and
sub
trac
tion
fact
s ne
ed to
be
reca
lled
auto
mat
ical
ly.
Rese
arch
cle
arly
indi
cate
s th
at th
ere
are
mor
e ef
fect
ive
way
s to
do
this
than
“dril
l an
d pr
actic
e”. A
bove
all,
it h
elps
to h
ave
stra
tegi
es fo
r wor
king
them
out
. Cou
ntin
g on
, usi
ng d
oubl
es a
nd u
sing
10s
are
goo
d st
rate
gies
, alth
ough
lear
ners
freq
uent
ly
inve
nt m
etho
ds th
at w
ork
equa
lly w
ell f
or
them
selv
es.
Diff
icul
ties
with
frac
tions
can
aris
e w
hen
frac
tiona
l not
atio
n is
intr
oduc
ed b
efor
e st
uden
ts h
ave
fully
con
stru
cted
mea
ning
ab
out f
ract
ion
conc
epts
.
Not
esM
odel
ling
usin
g m
anip
ulat
ives
pro
vide
s a
valu
able
sca
ffol
d fo
r con
stru
ctin
g m
eani
ng a
bout
mat
hem
atic
al
conc
epts
. The
re s
houl
d be
regu
lar
oppo
rtun
ities
for l
earn
ers
to w
ork
with
a ra
nge
of m
anip
ulat
ives
and
to
disc
uss
and
nego
tiate
thei
r dev
elop
ing
unde
rsta
ndin
gs w
ith o
ther
s.
***E
xam
ples
of r
elat
ed d
enom
inat
ors
incl
ude
halv
es, q
uart
ers
(four
ths)
and
ei
ghth
s. T
hese
can
be
mod
elle
d ea
sily
by
fold
ing
strip
s or
squ
ares
of p
aper
.
The
inte
rpre
tatio
n an
d m
eani
ng o
f re
mai
nder
s ca
n ca
use
diff
icul
ty fo
r so
me
lear
ners
. Thi
s is
esp
ecia
lly tr
ue if
ca
lcul
ator
s ar
e be
ing
used
. For
exa
mpl
e,
67 ÷
4 =
16.
75. T
his
can
also
be
show
n as
16¾
or 1
6 r3
. Lea
rner
s ne
ed p
ract
ice
in p
rodu
cing
app
ropr
iate
ans
wer
s w
hen
usin
g re
mai
nder
s. F
or e
xam
ple,
for a
sc
hool
trip
with
25
stud
ents
, onl
y bu
ses
that
car
ry 2
0 st
uden
ts a
re a
vaila
ble.
A
rem
aind
er c
ould
not
be
left
beh
ind,
so
anot
her b
us w
ould
be
requ
ired!
Calc
ulat
or s
kills
mus
t not
be
igno
red.
A
ll an
swer
s sh
ould
be
chec
ked
for t
heir
reas
onab
lene
ss.
By re
flect
ing
on a
nd re
cord
ing
thei
r fin
ding
s in
mat
hem
atic
s le
arni
ng lo
gs,
stud
ents
beg
in to
not
ice
patt
erns
in th
e nu
mbe
rs th
at w
ill fu
rthe
r dev
elop
thei
r un
ders
tand
ing.
Not
esIt
is n
ot p
ract
ical
to c
ontin
ue to
dev
elop
an
d us
e ba
se 1
0 m
ater
ials
bey
ond
1,00
0.
Lear
ners
sho
uld
have
litt
le d
iffic
ulty
in
exte
ndin
g th
e pl
ace
valu
e sy
stem
onc
e th
ey h
ave
unde
rsto
od th
e gr
oupi
ng
patt
ern
up to
1,0
00. T
here
are
a n
umbe
r of
web
site
s w
here
virt
ual m
anip
ulat
ives
ca
n be
util
ized
for w
orki
ng w
ith la
rger
nu
mbe
rs.
Estim
atio
n pl
ays
a ke
y ro
le in
che
ckin
g th
e fe
asib
ility
of a
nsw
ers.
The
met
hod
of m
ultip
lyin
g nu
mbe
rs a
nd ig
norin
g th
e de
cim
al p
oint
, the
n ad
just
ing
the
answ
er b
y co
untin
g de
cim
al p
lace
s, d
oes
not g
ive
the
lear
ner a
n un
ders
tand
ing
of
why
it is
don
e. A
pplic
atio
n of
pla
ce v
alue
kn
owle
dge
mus
t pre
cede
this
app
licat
ion
of p
atte
rn.
Mea
sure
men
t is
an e
xcel
lent
way
of
expl
orin
g th
e us
e of
frac
tions
and
de
cim
als
and
thei
r int
erch
ange
.
Stud
ents
sho
uld
be g
iven
man
y op
port
uniti
es to
dis
cove
r the
link
be
twee
n fr
actio
ns a
nd d
ivis
ion.
A th
orou
gh u
nder
stan
ding
of
mul
tiplic
atio
n, fa
ctor
s an
d la
rge
num
bers
is re
quire
d be
fore
wor
king
with
ex
pone
nts.
Mathematics scope and sequence26
Samples
Several examples of how schools are using the planner to facilitate mathematics inquiries have been developed and trialled by IB World Schools offering the PYP. These examples are included in the HTML version 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 where mathematics concepts are strongly evident. Please send planners to [email protected] for possible inclusion on this site.
Top Related