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Mathematics scope and sequence

Primary Years Programme

PYP110Printed in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire

Published February 2009

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© International Baccalaureate Organization 2009

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Primary Years Programme


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


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


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



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



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.



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).


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).


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).


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.


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.


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


collect, display and interpret data using simple graphs, for example, bar graphs, line graphs

identify, read and interpret range and scale on graphs


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.


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).


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.


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


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Learning continuums


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


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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


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


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


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


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


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


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


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


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


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


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


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


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


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


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.


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.

mathematics scope and sequence › pdfs › scopes and sequences › pyp... · 2017-10-24 ·...

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