Mathematics scope and sequence - Älmhults · PDF fileMathematics scope and sequence 1 Introduction to the PYP mathematics scope and sequence The information in this scope and sequence

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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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


Lear

nin

g c

on

tin

uu

m fo

r d

ata

han

dlin

g

Ph

ase

1 P

has

e 2

Ph

ase

3 P

has

e 4

Co

nce

ptu

al u

nd

erst

and

ing

sW

e co

llect

info

rmat

ion

to m

ake

sens

e of

th

e w

orld

aro

und

us.

Org

aniz

ing

obje

cts

and

even

ts h

elp

s us

to

solv

e p

rob

lem

s.

Even

ts in

dai

ly li

fe in

volv

e ch

ance

.

Co

nce

ptu

al u

nd

erst

and

ing

sIn

form

atio

n ca

n b

e ex

pre

ssed

as

orga

nize

d an

d st

ruct

ured

dat

a.

Ob

ject

s an

d ev

ents

can

be

orga

nize

d in

d

iffe

rent

way

s.

Som

e ev

ents

in d

aily

life

are

mor

e lik

ely

to

hap

pen

than

oth

ers.

Co

nce

ptu

al u

nd

erst

and

ing

sD

ata

can

be

colle

cted

, org

aniz

ed,

dis

pla

yed

and

anal

ysed

in d

iffe

rent

way

s.

Dif

fere

nt g

rap

h fo

rms

high

light

dif

fere

nt

asp

ects

of d

ata

mor

e ef

ficie

ntly

.

Prob

abili

ty c

an b

e b

ased

on

exp

erim

enta

l ev

ents

in d

aily

life

.

Prob

abili

ty c

an b

e ex

pre

ssed

in n

umer

ical

no

tati

ons.

Co

nce

ptu

al u

nd

erst

and

ing

sD

ata

can

be

pre

sent

ed e

ffec

tive

ly fo

r val

id

inte

rpre

tati

on a

nd c

omm

unic

atio

n.

Rang

e, m

ode,

med

ian

and

mea

n ca

n b

e us

ed to

ana

lyse

sta

tisti

cal d

ata.

Prob

abili

ty c

an b

e re

pre

sent

ed o

n a

scal

e b

etw

een

0–1

or 0

%–1

00%

.

The

pro

bab

ility

of a

n ev

ent c

an b

e p

red

icte

d th

eore

tica

lly.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

set

s ca

n b

e or

gani

zed

•b

y d

iffe

rent

att

rib

utes

und

erst

and

that

info

rmat

ion

abou

t •

them

selv

es a

nd th

eir s

urro

und

ings

ca

n b

e ob

tain

ed in

dif

fere

nt w

ays

dis

cuss

cha

nce

in d

aily

eve

nts

•(im

pos

sib

le, m

ayb

e, c

erta

in).

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

set

s ca

n b

e or

gani

zed

•b

y on

e or

mor

e at

trib

utes

und

erst

and

that

info

rmat

ion

abou

t •

them

selv

es a

nd th

eir s

urro

und

ings

ca

n b

e co

llect

ed a

nd re

cord

ed in

d

iffe

rent

way

s

und

erst

and

the

conc

ept o

f cha

nce

in

•d

aily

eve

nts

(imp

ossi

ble

, les

s lik

ely,

m

ayb

e, m

ost l

ikel

y, c

erta

in).

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

dat

a ca

n b

e •

colle

cted

, dis

pla

yed

and

inte

rpre

ted

usin

g si

mp

le g

rap

hs, f

or e

xam

ple

, bar

gr

aphs

, lin

e gr

aphs

und

erst

and

that

sca

le c

an re

pre

sent

•

dif

fere

nt q

uant

itie

s in

gra

phs

und

erst

and

that

the

mod

e ca

n b

e •

used

to s

umm

ariz

e a

set o

f dat

a

und

erst

and

that

one

of t

he p

urp

oses

•

of a

dat

abas

e is

to a

nsw

er q

uest

ions

an

d so

lve

pro

ble

ms

und

erst

and

that

pro

bab

ility

is b

ased

•

on e

xper

imen

tal e

vent

s.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

dif

fere

nt t

ypes

of

•gr

aphs

hav

e sp

ecia

l pur

pos

es

und

erst

and

that

the

mod

e, m

edia

n,

•m

ean

and

rang

e ca

n su

mm

ariz

e a

set

of d

ata

und

erst

and

that

pro

bab

ility

can

be

•ex

pre

ssed

in s

cale

(0–1

) or p

er c

ent

(0%

–100

%)

und

erst

and

the

dif

fere

nce

bet

wee

n •

exp

erim

enta

l and

theo

reti

cal

pro

bab

ility

.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

rep

rese

nt in

form

atio

n th

roug

h •

pic

togr

aphs

and

tally

mar

ks

sort

and

lab

el re

al o

bje

cts

by

•at

trib

utes

.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

colle

ct a

nd re

pre

sent

dat

a in

dif

fere

nt

•ty

pes

of g

rap

hs, f

or e

xam

ple

, tal

ly

mar

ks, b

ar g

rap

hs

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

colle

ct, d

isp

lay

and

inte

rpre

t dat

a •

usin

g si

mp

le g

rap

hs, f

or e

xam

ple

, bar

gr

aphs

, lin

e gr

aphs

iden

tify

, rea

d an

d in

terp

ret r

ang

e an

d •

scal

e on

gra

phs

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

colle

ct, d

ispl

ay a

nd in

terp

ret d

ata

in

•ci

rcle

gra

phs

(pie

cha

rts)

and

line

gra

phs

iden

tify

, des

crib

e an

d ex

pla

in th

e •

rang

e, m

ode,

med

ian

and

mea

n in

a

set o

f dat

a

Learning continuums


repr

esen

t the

rela

tions

hip

bet

wee

n •

obje

cts

in s

ets

usin

g tr

ee, V

enn

and

Car

roll

diag

ram

s

exp

ress

the

chan

ce o

f an

even

t •

hap

pen

ing

usin

g w

ords

or p

hras

es

(imp

ossi

ble

, les

s lik

ely,

may

be,

mos

t lik

ely,

cer

tain

).

iden

tify

the

mod

e of

a s

et o

f dat

a•

use

tree

dia

gram

s to

exp

ress

•

pro

bab

ility

usi

ng s

imp

le fr

acti

ons.

set u

p a

sp

read

shee

t usi

ng s

imp

le

•fo

rmul

as to

man

ipul

ate

dat

a an

d to

cr

eate

gra

phs

exp

ress

pro

bab

iliti

es u

sing

sca

le (0

–1)

•or

per

cen

t (0%

–100

%).

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

crea

te p

icto

grap

hs a

nd ta

lly m

arks

•

crea

te li

ving

gra

phs

usi

ng re

al o

bje

cts

•an

d p

eop

le*

des

crib

e re

al o

bje

cts

and

even

ts b

y •

attr

ibut

es.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

colle

ct, d

isp

lay

and

inte

rpre

t dat

a fo

r •

the

pur

pos

e of

ans

wer

ing

ques

tion

s

crea

te a

pic

togr

aph

and

sam

ple

bar

•

grap

h of

real

ob

ject

s an

d in

terp

ret

dat

a b

y co

mp

arin

g qu

anti

ties

(for

ex

amp

le, m

ore,

few

er, l

ess

than

, gr

eate

r tha

n)

use

tree

, Ven

n an

d C

arro

ll d

iagr

ams

to

•ex

plo

re re

lati

onsh

ips

bet

wee

n d

ata

iden

tify

and

des

crib

e ch

ance

in d

aily

•

even

ts (i

mp

ossi

ble

, les

s lik

ely,

may

be,

m

ost l

ikel

y, c

erta

in).

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

des

ign

a su

rvey

and

sys

tem

atic

ally

•

colle

ct, o

rgan

ize

and

dis

pla

y d

ata

in

pic

togr

aphs

and

bar

gra

phs

sele

ct a

pp

rop

riat

e gr

aph

form

(s) t

o •

dis

pla

y d

ata

inte

rpre

t ran

ge

and

scal

e on

gra

phs

•

use

pro

bab

ility

to d

eter

min

e •

mat

hem

atic

ally

fair

and

unfa

ir ga

mes

an

d to

exp

lain

pos

sib

le o

utco

mes

exp

ress

pro

bab

ility

usi

ng s

imp

le

•fr

acti

ons.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

des

ign

a su

rvey

and

sys

tem

atic

ally

•

colle

ct, r

ecor

d, o

rgan

ize

and

dis

pla

y th

e d

ata

in a

bar

gra

ph,

circ

le g

rap

h,

line

grap

h

iden

tify

, des

crib

e an

d ex

pla

in th

e •

rang

e, m

ode,

med

ian

and

mea

n in

a

set o

f dat

a

crea

te a

nd m

anip

ulat

e an

ele

ctro

nic

•d

atab

ase

for t

heir

own

pur

pos

es

det

erm

ine

the

theo

reti

cal p

rob

abili

ty

•of

an

even

t and

exp

lain

why

it m

ight

d

iffe

r fro

m e

xper

imen

tal p

rob

abili

ty.

No

tes

Uni

ts o

f inq

uiry

will

be

rich

in

opp

ortu

niti

es fo

r col

lect

ing

and

orga

nizi

ng in

form

atio

n. It

may

be

usef

ul

for t

he te

ache

r to

pro

vid

e sc

affo

lds,

suc

h as

que

stio

ns fo

r exp

lora

tion

, and

the

mod

ellin

g of

gra

phs

and

dia

gram

s.

*Liv

ing

grap

hs re

fer t

o d

ata

that

is

orga

nize

d b

y p

hysi

cally

mov

ing

and

arra

ngin

g st

uden

ts o

r act

ual m

ater

ials

in

suc

h a

way

as

to s

how

and

com

par

e qu

anti

ties

.

No

tes

An

incr

easi

ng n

umb

er o

f com

put

er a

nd

web

-bas

ed a

pp

licat

ions

are

ava

ilab

le th

at

enab

le le

arne

rs to

man

ipul

ate

dat

a in

or

der

to c

reat

e gr

aphs

.

Stud

ents

sho

uld

have

a lo

t of e

xper

ienc

e of

org

aniz

ing

dat

a in

a v

arie

ty o

f way

s,

and

of ta

lkin

g ab

out t

he a

dva

ntag

es a

nd

dis

adva

ntag

es o

f eac

h. In

terp

reta

tion

s of

d

ata

shou

ld in

clud

e th

e in

form

atio

n th

at

cann

ot b

e co

nclu

ded

as

wel

l as

that

whi

ch

can.

It is

imp

orta

nt to

rem

emb

er th

at

the

chos

en fo

rmat

sho

uld

illus

trat

e th

e in

form

atio

n w

itho

ut b

ias.

No

tes

Usi

ng d

ata

that

has

bee

n co

llect

ed a

nd

save

d is

a s

imp

le w

ay to

beg

in d

iscu

ssin

g th

e m

ode.

A fu

rthe

r ext

ensi

on o

f mod

e is

to

form

ulat

e th

eori

es a

bou

t why

a c

erta

in

choi

ce is

the

mod

e.

Stud

ents

sho

uld

have

the

opp

ortu

nity

to

use

dat

abas

es, i

dea

lly, t

hose

cre

ated

us

ing

dat

a co

llect

ed b

y th

e st

uden

ts th

en

ente

red

into

a d

atab

ase

by

the

teac

her o

r to

get

her.

No

tes

A d

atab

ase

is a

col

lect

ion

of d

ata,

whe

re

the

dat

a ca

n b

e d

isp

laye

d in

man

y fo

rms.

Th

e d

ata

can

be

chan

ged

at a

ny ti

me.

A

spre

adsh

eet i

s a

typ

e of

dat

abas

e w

here

in

form

atio

n is

set

out

in a

tab

le. U

sing

a

com

mon

set

of d

ata

is a

goo

d w

ay fo

r st

uden

ts to

sta

rt to

set

up

thei

r ow

n d

atab

ases

. A u

nit o

f inq

uiry

wou

ld b

e an

exc

elle

nt s

ourc

e of

com

mon

dat

a fo

r st

uden

t pra

ctic

e.

Learning continuums


Very

you

ng c

hild

ren

view

the

wor

ld a

s a

pla

ce o

f pos

sib

iliti

es. T

he te

ache

r sho

uld

try

to in

trod

uce

pra

ctic

al e

xam

ple

s an

d sh

ould

use

ap

pro

pri

ate

voca

bul

ary.

D

iscu

ssio

ns a

bou

t cha

nce

in d

aily

eve

nts

shou

ld b

e re

leva

nt to

the

cont

ext o

f the

le

arne

rs.

Situ

atio

ns th

at c

ome

up n

atur

ally

in

the

clas

sroo

m, o

ften

thro

ugh

liter

atur

e,

pre

sent

op

por

tuni

ties

for d

iscu

ssin

g p

rob

abili

ty. D

iscu

ssio

ns n

eed

to ta

ke

pla

ce in

whi

ch s

tud

ents

can

sha

re th

eir

sens

e of

like

lihoo

d in

term

s th

at a

re u

sefu

l to

them

.

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

pre

sent

op

por

tuni

ties

for s

tud

ents

to

furt

her d

evel

op th

eir u

nder

stan

din

g of

st

atis

tics

and

pro

bab

ility

con

cep

ts.

Tech

nolo

gy

give

s us

the

opti

on o

f cr

eati

ng a

gra

ph

at th

e p

ress

of a

key

. Be

ing

able

to g

ener

ate

dif

fere

nt t

ypes

of

gra

phs

allo

ws

lear

ners

to e

xplo

re a

nd

app

reci

ate

the

attr

ibut

es o

f eac

h ty

pe

of

grap

h an

d it

s ef

ficac

y in

dis

pla

ying

the

dat

a.

Tech

nolo

gy

also

giv

es u

s th

e p

ossi

bili

ty

of ra

pid

ly re

plic

atin

g ra

ndom

eve

nts.

C

omp

uter

and

web

-bas

ed a

pp

licat

ions

ca

n b

e us

ed to

toss

coi

ns, r

oll d

ice,

and

ta

bul

ate

and

grap

h th

e re

sult

s.

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


Lear

nin

g c

on

tin

uu

m fo

r m

easu

rem

ent

Ph

ase

1P

has

e 2

Ph

ase

3P

has

e 4

Co

nce

ptu

al u

nd

erst

and

ing

sM

easu

rem

ent i

nvol

ves

com

par

ing

obje

cts

and

even

ts.

Ob

ject

s ha

ve a

ttri

but

es th

at c

an b

e m

easu

red

usin

g no

n-s

tand

ard

unit

s.

Even

ts c

an b

e or

der

ed a

nd s

eque

nced

.

Co

nce

ptu

al u

nd

erst

and

ing

sSt

and

ard

unit

s al

low

us

to h

ave

a co

mm

on

lang

uag

e to

iden

tify

, com

par

e, o

rder

and

se

quen

ce o

bje

cts

and

even

ts.

We

use

tool

s to

mea

sure

the

attr

ibut

es o

f ob

ject

s an

d ev

ents

.

Esti

mat

ion

allo

ws

us to

mea

sure

wit

h d

iffe

rent

leve

ls o

f acc

urac

y.

Co

nce

ptu

al u

nd

erst

and

ing

sO

bje

cts

and

even

ts h

ave

attr

ibut

es th

at

can

be

mea

sure

d us

ing

app

rop

riat

e to

ols.

Rela

tion

ship

s ex

ist b

etw

een

stan

dar

d un

its

that

mea

sure

the

sam

e at

trib

utes

.

Co

nce

ptu

al u

nd

erst

and

ing

sA

ccur

acy

of m

easu

rem

ents

dep

ends

on

the

situ

atio

n an

d th

e p

reci

sion

of t

he to

ol.

Con

vers

ion

of u

nits

and

mea

sure

men

ts

allo

ws

us to

mak

e se

nse

of th

e w

orld

we

live

in.

A ra

nge

of p

roce

dure

s ex

ists

to m

easu

re

dif

fere

nt a

ttri

but

es o

f ob

ject

s an

d ev

ents

.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

att

rib

utes

of r

eal

•ob

ject

s ca

n b

e co

mp

ared

and

d

escr

ibed

, for

exa

mp

le, l

ong

er,

shor

ter,

heav

ier,

emp

ty, f

ull,

hott

er,

cold

er

und

erst

and

that

eve

nts

in d

aily

•

rout

ines

can

be

des

crib

ed a

nd

sequ

ence

d, f

or e

xam

ple

, bef

ore,

aft

er,

bed

tim

e, s

tory

tim

e, to

day

, tom

orro

w.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

the

use

of s

tand

ard

unit

s •

to m

easu

re, f

or e

xam

ple

, len

gth

, mas

s,

mon

ey, t

ime,

tem

per

atur

e

und

erst

and

that

tool

s ca

n b

e us

ed to

•

mea

sure

und

erst

and

that

cal

end

ars

can

be

•us

ed to

det

erm

ine

the

dat

e, a

nd to

id

enti

fy a

nd s

eque

nce

day

s of

the

wee

k an

d m

onth

s of

the

year

und

erst

and

that

tim

e is

mea

sure

d •

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

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

the

use

of s

tand

ard

•un

its

to m

easu

re p

erim

eter

, are

a an

d vo

lum

e

und

erst

and

that

mea

sure

s ca

n fa

ll •

bet

wee

n nu

mb

ers

on a

mea

sure

men

t sc

ale,

for e

xam

ple

, 3½

kg,

bet

wee

n 4

cm a

nd 5

cm

und

erst

and

rela

tion

ship

s b

etw

een

•un

its,

for e

xam

ple

, met

res,

ce

ntim

etre

s an

d m

illim

etre

s

und

erst

and

an a

ngle

as

a m

easu

re o

f •

rota

tion

.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

pro

cedu

res

for f

ind

ing

•ar

ea, p

erim

eter

and

vol

ume

und

erst

and

the

rela

tion

ship

s b

etw

een

•ar

ea a

nd p

erim

eter

, bet

wee

n ar

ea a

nd

volu

me,

and

bet

wee

n vo

lum

e an

d ca

pac

ity

und

erst

and

unit

con

vers

ions

wit

hin

•m

easu

rem

ent s

yste

ms

(met

ric

or

cust

omar

y).

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

iden

tify

, com

par

e an

d d

escr

ibe

•at

trib

utes

of r

eal o

bje

cts,

for e

xam

ple

, lo

nger

, sho

rter

, hea

vier

, em

pty

, ful

l, ho

tter

, col

der

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

esti

mat

e an

d m

easu

re o

bje

cts

usin

g •

stan

dar

d un

its

of m

easu

rem

ent:

leng

th, m

ass,

cap

acit

y, m

oney

and

te

mp

erat

ure

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

esti

mat

e an

d m

easu

re u

sing

sta

ndar

d •

unit

s of

mea

sure

men

t: p

erim

eter

, are

a an

d vo

lum

e

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

dev

elop

and

des

crib

e fo

rmul

as fo

r •

find

ing

per

imet

er, a

rea

and

volu

me

use

dec

imal

and

frac

tion

not

atio

n in

•

mea

sure

men

t, fo

r exa

mp

le, 3

.2 c

m,

1.47

kg,

1½

mile

s

Learning continuums


com

par

e th

e le

ngth

, mas

s an

d •

cap

acit

y of

ob

ject

s us

ing

non

-st

and

ard

unit

s

iden

tify

, des

crib

e an

d se

quen

ce

•ev

ents

in th

eir d

aily

rout

ine,

for

exam

ple

, bef

ore,

aft

er, b

edti

me,

st

oryt

ime,

tod

ay, t

omor

row

.

read

and

wri

te th

e ti

me

to th

e ho

ur,

•ha

lf ho

ur a

nd q

uart

er h

our

esti

mat

e an

d co

mp

are

leng

ths

of ti

me:

•

seco

nd, m

inut

e, h

our,

day

, wee

k an

d m

onth

.

des

crib

e m

easu

res

that

fall

bet

wee

n •

num

ber

s on

a s

cale

read

and

wri

te d

igit

al a

nd a

nalo

gue

•ti

me

on 1

2-ho

ur a

nd 2

4-ho

ur c

lock

s.

read

and

inte

rpre

t sca

les

on a

rang

e of

•

mea

suri

ng in

stru

men

ts

mea

sure

and

con

stru

ct a

ngle

s in

•

deg

rees

usi

ng a

pro

trac

tor

carr

y ou

t sim

ple

uni

t con

vers

ions

•

wit

hin

a sy

stem

of m

easu

rem

ent

(met

ric

or c

usto

mar

y).

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

des

crib

e ob

serv

atio

ns a

bou

t eve

nts

•an

d ob

ject

s in

real

-life

sit

uati

ons

use

non

-sta

ndar

d un

its

of

•m

easu

rem

ent t

o so

lve

pro

ble

ms

in

real

-life

sit

uati

ons

invo

lvin

g le

ngth

, m

ass

and

cap

acit

y.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

use

stan

dar

d un

its

of m

easu

rem

ent t

o •

solv

e p

rob

lem

s in

real

-life

sit

uati

ons

invo

lvin

g le

ngth

, mas

s, c

apac

ity,

m

oney

and

tem

per

atur

e

use

mea

sure

s of

tim

e to

ass

ist w

ith

•p

rob

lem

sol

ving

in re

al-li

fe s

itua

tion

s.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

use

stan

dar

d un

its

of m

easu

rem

ent t

o •

solv

e p

rob

lem

s in

real

-life

sit

uati

ons

invo

lvin

g p

erim

eter

, are

a an

d vo

lum

e

sele

ct a

pp

rop

riat

e to

ols

and

unit

s of

•

mea

sure

men

t

use

tim

elin

es in

uni

ts o

f inq

uiry

and

•

othe

r rea

l-lif

e si

tuat

ions

.

Whe

n •

app

lyin

g w

ith

un

der

stan

din

g

lear

ners

:

sele

ct a

nd u

se a

pp

rop

riat

e un

its

•of

mea

sure

men

t and

tool

s to

sol

ve

pro

ble

ms

in re

al-li

fe s

itua

tion

s

det

erm

ine

and

just

ify

the

leve

l of

•ac

cura

cy re

quire

d to

sol

ve re

al-li

fe

pro

ble

ms

invo

lvin

g m

easu

rem

ent

use

dec

imal

and

frac

tion

al n

otat

ion

•in

mea

sure

men

t, fo

r exa

mp

le, 3

.2 c

m,

1.47

kg,

1½

mile

s

use

tim

etab

les

and

sche

dule

s (1

2-•

hour

and

24-

hour

clo

cks)

in re

al-li

fe

situ

atio

ns

det

erm

ine

tim

es w

orld

wid

e.•

Learning continuums


No

tes

Lear

ners

nee

d m

any

opp

ortu

niti

es to

ex

per

ienc

e an

d qu

anti

fy m

easu

rem

ent

in a

dire

ct k

ines

thet

ic m

anne

r. Th

ey w

ill

dev

elop

und

erst

and

ing

of m

easu

rem

ent

by

usin

g m

anip

ulat

ives

and

mat

eria

ls

from

thei

r im

med

iate

env

ironm

ent,

for

exam

ple

, con

tain

ers

of d

iffe

rent

siz

es,

sand

, wat

er, b

eads

, cor

ks a

nd b

eans

.

No

tes

Usi

ng m

ater

ials

from

thei

r im

med

iate

en

viro

nmen

t, le

arne

rs c

an in

vest

igat

e ho

w u

nits

are

use

d fo

r mea

sure

men

t an

d ho

w m

easu

rem

ents

var

y d

epen

din

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 b

y b

asin

g es

tim

atio

ns o

n p

rior

kn

owle

dg

e, m

easu

ring

the

obje

ct a

nd

com

par

ing

actu

al m

easu

rem

ents

wit

h th

eir e

stim

atio

ns.

No

tes

In o

rder

to u

se m

easu

rem

ent m

ore

auth

enti

cally

, lea

rner

s sh

ould

hav

e th

e op

por

tuni

ty to

mea

sure

real

ob

ject

s in

re

al s

itua

tion

s. T

he u

nits

of i

nqui

ry c

an

ofte

n p

rovi

de

thes

e re

alis

tic

cont

exts

.

A w

ide

rang

e of

mea

suri

ng to

ols

shou

ld

be

avai

lab

le to

the

stud

ents

, for

exa

mp

le,

rule

rs, t

rund

le w

heel

s, ta

pe

mea

sure

s,

bat

hroo

m s

cale

s, k

itche

n sc

ales

, ti

mer

s, a

nalo

gue

cloc

ks, d

igit

al c

lock

s,

stop

wat

ches

and

cal

end

ars.

The

re a

re a

n in

crea

sing

num

ber

of c

omp

uter

and

web

-b

ased

ap

plic

atio

ns a

vaila

ble

for s

tud

ents

to

use

in a

uthe

ntic

con

text

s.

Plea

se n

ote

that

out

com

es re

lati

ng to

an

gles

als

o ap

pea

r in

the

shap

e an

d sp

ace

stra

nd.

No

tes

Lear

ners

gen

eral

ize

thei

r mea

suri

ng

exp

erie

nces

as

they

dev

ise

pro

cedu

res

and

form

ulas

for w

orki

ng o

ut p

erim

eter

, ar

ea a

nd v

olum

e.

Whi

le th

e em

pha

sis

for u

nder

stan

din

g 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

b

eing

aw

are

of th

e ex

iste

nce

of o

ther

sy

stem

s an

d ho

w c

onve

rsio

ns b

etw

een

syst

ems

help

us

to m

ake

sens

e of

them

.

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

nin

g c

on

tin

uu

m fo

r sh

ape

and

sp

ace

Ph

ase

1P

has

e 2

Ph

ase

3P

has

e 4

Co

nce

ptu

al u

nd

erst

and

ing

sSh

apes

can

be

des

crib

ed a

nd o

rgan

ized

ac

cord

ing

to th

eir p

rop

erti

es.

Ob

ject

s in

our

imm

edia

te e

nviro

nmen

t ha

ve a

pos

itio

n in

sp

ace

that

can

be

des

crib

ed a

ccor

din

g to

a p

oint

of

refe

renc

e.

Co

nce

ptu

al u

nd

erst

and

ing

sSh

apes

are

cla

ssifi

ed a

nd n

amed

ac

cord

ing

to th

eir p

rop

erti

es.

Som

e sh

apes

are

mad

e up

of p

arts

that

re

pea

t in

som

e w

ay.

Spec

ific

voca

bul

ary

can

be

used

to

des

crib

e an

ob

ject

’s p

osit

ion

in s

pac

e.

Co

nce

ptu

al u

nd

erst

and

ing

sC

hang

ing

the

pos

itio

n of

a s

hap

e d

oes

not a

lter

its

pro

per

ties

.

Shap

es c

an b

e tr

ansf

orm

ed in

dif

fere

nt

way

s.

Geo

met

ric

shap

es a

nd v

ocab

ular

y ar

e us

eful

for r

epre

sent

ing

and

des

crib

ing

obje

cts

and

even

ts in

real

-wor

ld

situ

atio

ns.

Co

nce

ptu

al u

nd

erst

and

ing

sM

anip

ulat

ion

of s

hap

e an

d sp

ace

take

s p

lace

for a

par

ticu

lar p

urp

ose.

Con

solid

atin

g w

hat w

e kn

ow o

f g

eom

etri

c co

ncep

ts a

llow

s us

to m

ake

sens

e of

and

inte

ract

wit

h ou

r wor

ld.

Geo

met

ric

tool

s an

d m

etho

ds c

an b

e us

ed to

sol

ve p

rob

lem

s re

lati

ng to

sha

pe

and

spac

e.

Lear

nin

g o

utc

om

es

Whe

n co

nst

ruct

ing

mea

nin

g le

arne

rs:

und

erst

and

that

2D

and

3D

sha

pes

•

have

cha

ract

eris

tics

that

can

be

des

crib

ed a

nd c

omp

ared

und

erst

and

that

com

mon

lang

uag

e •

can

be

used

to d

escr

ibe

pos

itio

n an

d d

irect

ion,

for e

xam

ple

, ins

ide,

out

sid

e,

abov

e, b

elow

, nex

t to,

beh

ind

, in

fron

t of

, up,

dow

n.

Lear

nin

g o

utc

om

es

Whe

n co

nst

ruct

ing

mea

nin

g le

arne

rs:

und

erst

and

that

ther

e ar

e •

rela

tion

ship

s am

ong

and

bet

wee

n 2D

an

d 3D

sha

pes

und

erst

and

that

2D

and

3D

sha

pes

•

can

be

crea

ted

by

put

ting

tog

ethe

r an

d/o

r tak

ing

apar

t oth

er s

hap

es

und

erst

and

that

exa

mp

les

of

•sy

mm

etry

and

tran

sfor

mat

ions

can

be

foun

d in

thei

r im

med

iate

env

ironm

ent

und

erst

and

that

geo

met

ric

shap

es

•ar

e us

eful

for r

epre

sent

ing

real

-wor

ld

situ

atio

ns

und

erst

and

that

dire

ctio

ns c

an b

e •

used

to d

escr

ibe

pat

hway

s, re

gion

s,

pos

itio

ns a

nd b

ound

arie

s of

thei

r im

med

iate

env

ironm

ent.

Lear

nin

g o

utc

om

es

Whe

n co

nst

ruct

ing

mea

nin

g le

arne

rs:

und

erst

and

the

com

mon

lang

uag

e •

used

to d

escr

ibe

shap

es

und

erst

and

the

pro

per

ties

of r

egul

ar

•an

d ir

regu

lar p

olyg

ons

und

erst

and

cong

ruen

t or s

imila

r •

shap

es

und

erst

and

that

line

s an

d ax

es o

f •

refl

ecti

ve a

nd ro

tati

onal

sym

met

ry

assi

st w

ith

the

cons

truc

tion

of s

hap

es

und

erst

and

an a

ngle

as

a m

easu

re o

f •

rota

tion

und

erst

and

that

dire

ctio

ns fo

r •

loca

tion

can

be

rep

rese

nted

by

coor

din

ates

on

a gr

id

und

erst

and

that

vis

ualiz

atio

n of

sha

pe

•an

d sp

ace

is a

str

ateg

y fo

r sol

ving

p

rob

lem

s.

Lear

nin

g o

utc

om

es

Whe

n co

nst

ruct

ing

mea

nin

g le

arne

rs:

und

erst

and

the

com

mon

lang

uag

e •

used

to d

escr

ibe

shap

es

und

erst

and

the

pro

per

ties

of r

egul

ar

•an

d ir

regu

lar p

olyh

edra

und

erst

and

the

pro

per

ties

of c

ircle

s•

und

erst

and

how

sca

le (r

atio

s) is

use

d •

to e

nlar

ge

and

redu

ce s

hap

es

und

erst

and

syst

ems

for d

escr

ibin

g •

pos

itio

n an

d d

irect

ion

und

erst

and

that

2D

rep

rese

ntat

ions

•

of 3

D o

bje

cts

can

be

used

to v

isua

lize

and

solv

e p

rob

lem

s

und

erst

and

that

geo

met

ric

idea

s •

and

rela

tion

ship

s ca

n b

e us

ed to

so

lve

pro

ble

ms

in o

ther

are

as o

f m

athe

mat

ics

and

in re

al li

fe.

Learning continuums


Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

sort

, des

crib

e an

d co

mp

are

3D s

hap

es

•

des

crib

e p

osit

ion

and

dire

ctio

n, fo

r •

exam

ple

, ins

ide,

out

sid

e, a

bov

e,

bel

ow, n

ext t

o, b

ehin

d, i

n fr

ont o

f, up

, d

own.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

sort

, des

crib

e an

d la

bel

2D

and

3D

•

shap

es

anal

yse

and

des

crib

e th

e re

lati

onsh

ips

•b

etw

een

2D a

nd 3

D s

hap

es

crea

te a

nd d

escr

ibe

sym

met

rica

l and

•

tess

ella

ting

pat

tern

s

iden

tify

line

s of

refl

ecti

ve s

ymm

etry

•

rep

rese

nt id

eas

abou

t the

real

wor

ld

•us

ing

geo

met

ric

voca

bul

ary

and

sym

bol

s, fo

r exa

mp

le, t

hrou

gh o

ral

des

crip

tion

, dra

win

g, m

odel

ling,

la

bel

ling

inte

rpre

t and

cre

ate

sim

ple

dire

ctio

ns,

•d

escr

ibin

g p

aths

, reg

ions

, pos

itio

ns

and

bou

ndar

ies

of th

eir i

mm

edia

te

envi

ronm

ent.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

sort

, des

crib

e an

d m

odel

regu

lar a

nd

•ir

regu

lar p

olyg

ons

des

crib

e an

d m

odel

con

grue

ncy

and

•si

mila

rity

in 2

D s

hap

es

anal

yse

angl

es b

y co

mp

arin

g an

d •

des

crib

ing

rota

tion

s: w

hole

turn

; hal

f tu

rn; q

uart

er tu

rn; n

orth

, sou

th, e

ast

and

wes

t on

a co

mp

ass

loca

te fe

atur

es o

n a

grid

usi

ng

•co

ord

inat

es

des

crib

e an

d/o

r rep

rese

nt m

enta

l •

imag

es o

f ob

ject

s, p

atte

rns,

and

p

aths

.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

anal

yse,

des

crib

e, c

lass

ify

and

visu

aliz

e •

2D (i

nclu

din

g ci

rcle

s, tr

iang

les

and

quad

rila

tera

ls) a

nd 3

D s

hap

es, u

sing

g

eom

etri

c vo

cab

ular

y

des

crib

e lin

es a

nd a

ngle

s us

ing

•g

eom

etri

c vo

cab

ular

y

iden

tify

and

use

sca

le (r

atio

s) to

•

enla

rge

and

redu

ce s

hap

es

iden

tify

and

use

the

lang

uag

e an

d •

nota

tion

of b

eari

ng to

des

crib

e d

irect

ion

and

pos

itio

n

crea

te a

nd m

odel

how

a 2

D n

et

•co

nver

ts in

to a

3D

sha

pe

and

vice

ve

rsa

exp

lore

the

use

of g

eom

etri

c id

eas

•an

d re

lati

onsh

ips

to s

olve

pro

ble

ms

in

othe

r are

as o

f mat

hem

atic

s.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

exp

lore

and

des

crib

e th

e p

aths

, •

regi

ons

and

bou

ndar

ies

of th

eir

imm

edia

te e

nviro

nmen

t (in

sid

e,

outs

ide,

ab

ove,

bel

ow) a

nd th

eir

pos

itio

n (n

ext t

o, b

ehin

d, i

n fr

ont o

f, up

, dow

n).

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

anal

yse

and

use

wha

t the

y kn

ow

•ab

out 3

D s

hap

es to

des

crib

e an

d w

ork

wit

h 2D

sha

pes

reco

gniz

e an

d ex

pla

in s

imp

le

•sy

mm

etri

cal d

esig

ns in

the

envi

ronm

ent

app

ly k

now

led

ge

of s

ymm

etry

to

•p

rob

lem

-sol

ving

sit

uati

ons

inte

rpre

t and

use

sim

ple

dire

ctio

ns,

•d

escr

ibin

g p

aths

, reg

ions

, pos

itio

ns

and

bou

ndar

ies

of th

eir i

mm

edia

te

envi

ronm

ent.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

anal

yse

and

des

crib

e 2D

and

3D

•

shap

es, i

nclu

din

g re

gula

r and

ir

regu

lar p

olyg

ons,

usi

ng g

eom

etri

cal

voca

bul

ary

iden

tify

, des

crib

e an

d m

odel

•

cong

ruen

cy a

nd s

imila

rity

in 2

D

shap

es

reco

gniz

e an

d ex

pla

in s

ymm

etri

cal

•p

atte

rns,

incl

udin

g te

ssel

lati

on, i

n th

e en

viro

nmen

t

app

ly k

now

led

ge

of tr

ansf

orm

atio

ns

•to

pro

ble

m-s

olvi

ng s

itua

tion

s.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

use

geo

met

ric

voca

bul

ary

whe

n •

des

crib

ing

shap

e an

d sp

ace

in

mat

hem

atic

al s

itua

tion

s an

d b

eyon

d

use

scal

e (r

atio

s) to

enl

arg

e an

d •

redu

ce s

hap

es

app

ly th

e la

ngua

ge

and

nota

tion

of

•b

eari

ng to

des

crib

e d

irect

ion

and

pos

itio

n

use

2D re

pre

sent

atio

ns o

f 3D

ob

ject

s •

to v

isua

lize

and

solv

e p

rob

lem

s, fo

r ex

amp

le u

sing

dra

win

gs o

r mod

els.

Learning continuums


No

tes

Lear

ners

nee

d m

any

opp

ortu

niti

es to

ex

per

ienc

e sh

ape

and

spac

e in

a d

irect

ki

nest

heti

c m

anne

r, fo

r exa

mp

le, t

hrou

gh

pla

y, 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

par

ticu

lar t

he re

al-li

fe o

bje

cts

wit

h w

hich

ch

ildre

n ar

e fa

mili

ar. 2

D s

hap

es (p

lane

sh

apes

) are

a m

ore

abst

ract

con

cep

t but

ca

n b

e un

der

stoo

d as

face

s of

3D

sha

pes

.

No

tes

Lear

ners

nee

d to

und

erst

and

the

pro

per

ties

of 2

D a

nd 3

D s

hap

es

bef

ore

the

mat

hem

atic

al v

ocab

ular

y as

soci

ated

wit

h sh

apes

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

mor

e fo

rmal

m

athe

mat

ical

voc

abul

ary

and

beg

in to

ap

pre

ciat

e th

e ne

ed fo

r thi

s p

reci

sion

.

No

tes

Com

put

er a

nd w

eb-b

ased

ap

plic

atio

ns

can

be

used

to e

xplo

re s

hap

e an

d sp

ace

conc

epts

suc

h as

sym

met

ry, a

ngle

s an

d co

ord

inat

es.

The

unit

s of

inqu

iry

can

pro

vid

e au

then

tic

cont

exts

for d

evel

opin

g un

der

stan

din

g of

con

cep

ts re

lati

ng to

loca

tion

and

d

irect

ions

.

No

tes

Tool

s su

ch a

s co

mp

asse

s an

d p

rotr

acto

rs

are

com

mon

ly u

sed

to s

olve

pro

ble

ms

in

real

-life

sit

uati

ons.

How

ever

, car

e sh

ould

b

e ta

ken

to e

nsur

e th

at s

tud

ents

hav

e a

stro

ng u

nder

stan

din

g of

the

conc

epts

em

bed

ded

in th

e p

rob

lem

to e

nsur

e m

eani

ngfu

l eng

agem

ent w

ith

the

tool

s an

d fu

ll un

der

stan

din

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

nin

g c

on

tin

uu

m fo

r p

atte

rn a

nd

fun

ctio

n

Ph

ase

1P

has

e 2

Ph

ase

3P

has

e 4

Co

nce

ptu

al u

nd

erst

and

ing

s

Patt

erns

and

seq

uenc

es o

ccur

in e

very

day

situ

atio

ns.

Patt

erns

rep

eat a

nd g

row

.

Co

nce

ptu

al u

nd

erst

and

ing

s

Who

le n

umb

ers

exhi

bit

pat

tern

s an

d

rela

tion

ship

s th

at c

an b

e ob

serv

ed a

nd

des

crib

ed.

Patt

erns

can

be

rep

rese

nted

usi

ng

num

ber

s an

d ot

her s

ymb

ols.

Co

nce

ptu

al u

nd

erst

and

ing

s

Func

tion

s ar

e re

lati

onsh

ips

or ru

les

that

uniq

uely

ass

ocia

te m

emb

ers

of o

ne s

et

wit

h m

emb

ers

of a

noth

er s

et.

By a

naly

sing

pat

tern

s an

d id

enti

fyin

g

rule

s fo

r pat

tern

s it

is p

ossi

ble

to m

ake

pre

dic

tion

s.

Co

nce

ptu

al u

nd

erst

and

ing

s

Patt

erns

can

oft

en b

e g

ener

aliz

ed u

sing

alg

ebra

ic e

xpre

ssio

ns, e

quat

ions

or

func

tion

s.

Exp

onen

tial

not

atio

n is

a p

ower

ful w

ay

to e

xpre

ss re

pea

ted

pro

duct

s of

the

sam

e

num

ber

.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

pat

tern

s ca

n b

e •

foun

d in

eve

ryd

ay s

itua

tion

s, fo

r ex

amp

le, s

ound

s, a

ctio

ns, o

bje

cts,

na

ture

.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

pat

tern

s ca

n b

e •

foun

d in

num

ber

s, fo

r exa

mp

le, o

dd

and

even

num

ber

s, s

kip

cou

ntin

g

und

erst

and

the

inve

rse

rela

tion

ship

•

bet

wee

n ad

dit

ion

and

sub

trac

tion

und

erst

and

the

asso

ciat

ive

and

•co

mm

utat

ive

pro

per

ties

of a

dd

itio

n.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

pat

tern

s ca

n b

e •

anal

ysed

and

rule

s id

enti

fied

und

erst

and

that

mul

tip

licat

ion

is

•re

pea

ted

add

itio

n an

d th

at d

ivis

ion

is

rep

eate

d su

btr

acti

on

und

erst

and

the

inve

rse

rela

tion

ship

•

bet

wee

n m

ulti

plic

atio

n an

d d

ivis

ion

und

erst

and

the

asso

ciat

ive

•an

d co

mm

utat

ive

pro

per

ties

of

mul

tip

licat

ion.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

that

pat

tern

s ca

n b

e •

gen

eral

ized

by

a ru

le

und

erst

and

exp

onen

ts a

s re

pea

ted

•m

ulti

plic

atio

n

und

erst

and

the

inve

rse

rela

tion

ship

•

bet

wee

n ex

pon

ents

and

root

s

und

erst

and

that

pat

tern

s ca

n b

e •

rep

rese

nted

, ana

lyse

d an

d g

ener

aliz

ed

usin

g ta

ble

s, g

rap

hs, w

ords

, and

, w

hen

pos

sib

le, s

ymb

olic

rule

s.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

des

crib

e p

atte

rns

in v

ario

us w

ays,

•

for e

xam

ple

, usi

ng w

ords

, dra

win

gs,

sym

bol

s, m

ater

ials

, act

ions

, num

ber

s.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

rep

rese

nt p

atte

rns

in a

var

iety

of w

ays,

•

for e

xam

ple

, usi

ng w

ords

, dra

win

gs,

sym

bol

s, m

ater

ials

, act

ions

, num

ber

s

des

crib

e nu

mb

er p

atte

rns,

for

•ex

amp

le, o

dd

and

even

num

ber

s, s

kip

co

unti

ng.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

des

crib

e th

e ru

le fo

r a p

atte

rn in

a

•va

riet

y of

way

s

rep

rese

nt ru

les

for p

atte

rns

usin

g •

wor

ds, s

ymb

ols

and

tab

les

iden

tify

a s

eque

nce

of o

per

atio

ns

•re

lati

ng o

ne s

et o

f num

ber

s to

ano

ther

se

t.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

rep

rese

nt th

e ru

le o

f a p

atte

rn b

y •

usin

g a

func

tion

anal

yse

pat

tern

and

func

tion

usi

ng

•w

ords

, tab

les

and

grap

hs, a

nd, w

hen

pos

sib

le, s

ymb

olic

rule

s.

Learning continuums


Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

exte

nd a

nd c

reat

e p

atte

rns.

•

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

exte

nd a

nd c

reat

e p

atte

rns

in

•nu

mb

ers,

for e

xam

ple

, od

d an

d ev

en

num

ber

s, s

kip

cou

ntin

g

use

num

ber

pat

tern

s to

rep

rese

nt a

nd

•un

der

stan

d re

al-li

fe s

itua

tion

s

use

the

pro

per

ties

and

rela

tion

ship

s •

of a

dd

itio

n an

d su

btr

acti

on to

sol

ve

pro

ble

ms.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

sele

ct a

pp

rop

riat

e m

etho

ds fo

r •

rep

rese

ntin

g p

atte

rns,

for e

xam

ple

us

ing

wor

ds, s

ymb

ols

and

tab

les

use

num

ber

pat

tern

s to

mak

e •

pre

dic

tion

s an

d so

lve

pro

ble

ms

use

the

pro

per

ties

and

rela

tion

ship

s of

•

the

four

op

erat

ions

to s

olve

pro

ble

ms.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

le

arne

rs:

sele

ct a

pp

rop

riat

e m

etho

ds to

ana

lyse

•

pat

tern

s an

d id

enti

fy ru

les

use

func

tion

s to

sol

ve p

rob

lem

s.•

No

tes

The

wor

ld is

fille

d w

ith

pat

tern

and

ther

e w

ill b

e m

any

opp

ortu

niti

es fo

r lea

rner

s to

mak

e th

is c

onne

ctio

n ac

ross

the

curr

icul

um.

A ra

nge

of m

anip

ulat

ives

can

be

used

to

exp

lore

pat

tern

s in

clud

ing

pat

tern

blo

cks,

at

trib

ute

blo

cks,

col

our t

iles,

cal

cula

tors

, nu

mb

er c

hart

s, b

eans

and

but

tons

.

No

tes

Stud

ents

will

ap

ply

thei

r und

erst

and

ing

of p

atte

rn to

the

num

ber

s th

ey a

lread

y kn

ow. T

he p

atte

rns

they

find

will

hel

p to

d

eep

en th

eir u

nder

stan

din

g of

a ra

nge

of

num

ber

con

cep

ts.

Four

-fun

ctio

n ca

lcul

ator

s ca

n b

e us

ed to

ex

plo

re n

umb

er p

atte

rns.

No

tes

Patt

erns

are

cen

tral

to th

e un

der

stan

din

g of

all

conc

epts

in m

athe

mat

ics.

The

y ar

e th

e b

asis

of h

ow o

ur n

umb

er s

yste

m is

or

gani

zed

. Sea

rchi

ng fo

r, an

d id

enti

fyin

g,

pat

tern

s he

lps

us to

see

rela

tion

ship

s,

mak

e g

ener

aliz

atio

ns, a

nd is

a p

ower

ful

stra

teg

y fo

r pro

ble

m s

olvi

ng. F

unct

ions

d

evel

op fr

om th

e st

udy

of p

atte

rns

and

mak

e it

pos

sib

le to

pre

dic

t in

mat

hem

atic

s p

rob

lem

s.

No

tes

Alg

ebra

is a

mat

hem

atic

al la

ngua

ge

usin

g nu

mb

ers

and

sym

bol

s to

exp

ress

re

lati

onsh

ips.

Whe

n th

e sa

me

rela

tion

ship

w

orks

wit

h an

y nu

mb

er, a

lgeb

ra u

ses

lett

ers

to re

pre

sent

the

gen

eral

izat

ion.

Le

tter

s ca

n b

e us

ed to

rep

rese

nt th

e qu

anti

ty.

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

nin

g c

on

tin

uu

m fo

r n

um

ber

Ph

ase

1P

has

e 2

Ph

ase

3P

has

e 4

Co

nce

ptu

al u

nd

erst

and

ing

sN

umb

ers

are

a na

min

g sy

stem

.

Num

ber

s ca

n b

e us

ed in

man

y w

ays

for

dif

fere

nt p

urp

oses

in th

e re

al w

orld

.

Num

ber

s ar

e co

nnec

ted

to e

ach

othe

r th

roug

h a

vari

ety

of re

lati

onsh

ips.

Mak

ing

conn

ecti

ons

bet

wee

n ou

r ex

per

ienc

es w

ith

num

ber

can

hel

p u

s to

d

evel

op n

umb

er s

ense

.

Co

nce

ptu

al u

nd

erst

and

ing

sTh

e b

ase

10 p

lace

val

ue s

yste

m is

use

d to

rep

rese

nt n

umb

ers

and

num

ber

re

lati

onsh

ips.

Frac

tion

s ar

e w

ays

of re

pre

sent

ing

who

le-

par

t rel

atio

nshi

ps.

The

oper

atio

ns o

f ad

dit

ion,

sub

trac

tion

, m

ulti

plic

atio

n an

d d

ivis

ion

are

rela

ted

to e

ach

othe

r and

are

use

d to

pro

cess

in

form

atio

n to

sol

ve p

rob

lem

s.

Num

ber

op

erat

ions

can

be

mod

elle

d in

a

vari

ety

of w

ays.

Ther

e ar

e m

any

men

tal m

etho

ds th

at c

an

be

app

lied

for e

xact

and

ap

pro

xim

ate

com

put

atio

ns.

Co

nce

ptu

al u

nd

erst

and

ing

sTh

e b

ase

10 p

lace

val

ue s

yste

m c

an b

e ex

tend

ed to

rep

rese

nt m

agni

tud

e.

Frac

tion

s an

d d

ecim

als

are

way

s of

re

pre

sent

ing

who

le-p

art r

elat

ions

hip

s.

The

oper

atio

ns o

f ad

dit

ion,

sub

trac

tion

, m

ulti

plic

atio

n an

d d

ivis

ion

are

rela

ted

to e

ach

othe

r and

are

use

d to

pro

cess

in

form

atio

n to

sol

ve p

rob

lem

s.

Even

com

ple

x op

erat

ions

can

be

mod

elle

d in

a v

arie

ty o

f way

s, fo

r ex

amp

le, a

n al

gor

ithm

is a

way

to

rep

rese

nt a

n op

erat

ion.

Co

nce

ptu

al u

nd

erst

and

ing

sTh

e b

ase

10 p

lace

val

ue s

yste

m e

xten

ds

infin

itely

in t

wo

dire

ctio

ns.

Frac

tion

s, d

ecim

al fr

acti

ons

and

per

cent

ages

are

way

s of

rep

rese

ntin

g w

hole

-par

t rel

atio

nshi

ps.

For f

ract

iona

l and

dec

imal

com

put

atio

n,

the

idea

s d

evel

oped

for w

hole

-num

ber

co

mp

utat

ion

can

app

ly.

Rati

os a

re a

com

par

ison

of t

wo

num

ber

s or

qua

ntit

ies.

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

und

erst

and

one-

to-o

ne

•co

rres

pon

den

ce

und

erst

and

that

, for

a s

et o

f ob

ject

s,

•th

e nu

mb

er n

ame

of th

e la

st o

bje

ct

coun

ted

des

crib

es th

e qu

anti

ty o

f the

w

hole

set

und

erst

and

that

num

ber

s ca

n •

be

cons

truc

ted

in m

ulti

ple

way

s,

for e

xam

ple

, by

com

bin

ing

and

par

titi

onin

g

und

erst

and

cons

erva

tion

of n

umb

er*

•

und

erst

and

the

rela

tive

mag

nitu

de

of

•w

hole

num

ber

s

reco

gniz

e gr

oup

s of

zer

o to

five

•

obje

cts

wit

hout

cou

ntin

g (s

ubit

izin

g)

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

mod

el n

umb

ers

to h

und

reds

or

•b

eyon

d us

ing

the

bas

e 10

pla

ce v

alue

sy

stem

**

esti

mat

e qu

anti

ties

to 1

00 o

r bey

ond

•

mod

el s

imp

le fr

acti

on re

lati

onsh

ips

•

use

the

lang

uag

e of

ad

dit

ion

and

•su

btr

acti

on, f

or e

xam

ple

, ad

d, t

ake

away

, plu

s, m

inus

, sum

, dif

fere

nce

mod

el a

dd

itio

n an

d su

btr

acti

on o

f •

who

le n

umb

ers

dev

elop

str

ateg

ies

for m

emor

izin

g •

add

itio

n an

d su

btr

acti

on n

umb

er

fact

s

esti

mat

e su

ms

and

dif

fere

nces

•

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

mod

el n

umb

ers

to th

ousa

nds

or

•b

eyon

d us

ing

the

bas

e 10

pla

ce v

alue

sy

stem

mod

el e

quiv

alen

t fra

ctio

ns•

use

the

lang

uag

e of

frac

tion

s, fo

r •

exam

ple

, num

erat

or, d

enom

inat

or

mod

el d

ecim

al fr

acti

ons

to

•hu

ndre

dth

s or

bey

ond

mod

el m

ulti

plic

atio

n an

d d

ivis

ion

of

•w

hole

num

ber

s

use

the

lang

uag

e of

mul

tip

licat

ion

•an

d d

ivis

ion,

for e

xam

ple

, fac

tor,

mul

tip

le, p

rodu

ct, q

uoti

ent,

pri

me

num

ber

s, c

omp

osite

num

ber

Lear

nin

g o

utc

om

esW

hen

con

stru

ctin

g m

ean

ing

lear

ners

:

mod

el n

umb

ers

to m

illio

ns o

r bey

ond

•us

ing

the

bas

e 10

pla

ce v

alue

sys

tem

mod

el ra

tios

•

mod

el in

teg

ers

in a

pp

rop

riat

e •

cont

exts

mod

el e

xpon

ents

and

squ

are

root

s •

mod

el im

pro

per

frac

tion

s an

d m

ixed

•

num

ber

s

sim

plif

y fr

acti

ons

usin

g m

anip

ulat

ives

•

mod

el d

ecim

al fr

acti

ons

to

•th

ousa

ndth

s or

bey

ond

mod

el p

erce

ntag

es•

und

erst

and

the

rela

tion

ship

bet

wee

n •

frac

tion

s, d

ecim

als

and

per

cent

ages

Learning continuums


und

erst

and

who

le-p

art r

elat

ions

hip

s •

use

the

lang

uag

e of

mat

hem

atic

s •

to c

omp

are

quan

titi

es, f

or e

xam

ple

, m

ore,

less

, fir

st, s

econ

d.

und

erst

and

situ

atio

ns th

at in

volv

e •

mul

tip

licat

ion

and

div

isio

n

mod

el a

dd

itio

n an

d su

btr

acti

on o

f •

frac

tion

s w

ith

the

sam

e d

enom

inat

or.

mod

el a

dd

itio

n an

d su

btr

acti

on

•of

frac

tion

s w

ith

rela

ted

den

omin

ator

s***

mod

el a

dd

itio

n an

d su

btr

acti

on o

f •

dec

imal

s.

mod

el a

dd

itio

n, s

ubtr

acti

on,

•m

ulti

plic

atio

n an

d d

ivis

ion

of fr

acti

ons

mod

el a

dd

itio

n, s

ubtr

acti

on,

•m

ulti

plic

atio

n an

d d

ivis

ion

of

dec

imal

s.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

conn

ect n

umb

er n

ames

and

•

num

eral

s to

the

quan

titie

s th

ey

rep

rese

nt.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

read

and

wri

te w

hole

num

ber

s up

to

•hu

ndre

ds o

r bey

ond

read

, wri

te, c

omp

are

and

ord

er

•ca

rdin

al a

nd o

rdin

al n

umb

ers

des

crib

e m

enta

l and

wri

tten

•

stra

tegi

es fo

r ad

din

g an

d su

btr

acti

ng

two

-dig

it n

umb

ers.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

read

, wri

te, c

omp

are

and

ord

er w

hole

•

num

ber

s up

to th

ousa

nds

or b

eyon

d

dev

elop

str

ateg

ies

for m

emor

izin

g •

add

itio

n, s

ubtr

acti

on, m

ulti

plic

atio

n an

d d

ivis

ion

num

ber

fact

s

read

, wri

te, c

omp

are

and

ord

er

•fr

acti

ons

read

and

wri

te e

quiv

alen

t fra

ctio

ns•

read

, wri

te, c

omp

are

and

ord

er

•fr

acti

ons

to h

und

red

ths

or b

eyon

d

des

crib

e m

enta

l and

wri

tten

•

stra

tegi

es fo

r mul

tip

licat

ion

and

div

isio

n.

Whe

n tr

ansf

erri

ng

mea

nin

g in

to

sym

bo

ls le

arne

rs:

read

, wri

te, c

omp

are

and

ord

er w

hole

•

num

ber

s up

to m

illio

ns o

r bey

ond

read

and

wri

te ra

tios

•

read

and

wri

te in

teg

ers

in a

pp

rop

riat

e •

cont

exts

read

and

wri

te e

xpon

ents

and

squ

are

•ro

ots

conv

ert i

mp

rop

er fr

acti

ons

to m

ixed

•

num

ber

s an

d vi

ce v

ersa

sim

plif

y fr

acti

ons

in m

enta

l and

•

wri

tten

form

read

, wri

te, c

omp

are

and

ord

er

•d

ecim

al fr

acti

ons

to th

ousa

ndth

s or

b

eyon

d

read

, wri

te, c

omp

are

and

ord

er

•p

erce

ntag

es

conv

ert b

etw

een

frac

tion

s, d

ecim

als

•an

d p

erce

ntag

es.

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

lear

ners

:

coun

t to

det

erm

ine

the

num

ber

of

•ob

ject

s in

a s

et

use

num

ber

wor

ds a

nd n

umer

als

•to

rep

rese

nt q

uant

itie

s in

real

-life

si

tuat

ions

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

lear

ners

:

use

who

le n

umb

ers

up to

hun

dre

ds o

r •

bey

ond

in re

al-li

fe s

itua

tion

s

use

card

inal

and

ord

inal

num

ber

s in

•

real

-life

sit

uati

ons

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

lear

ners

:

use

who

le n

umb

ers

up to

thou

sand

s •

or b

eyon

d in

real

-life

sit

uati

ons

use

fast

reca

ll of

mul

tip

licat

ion

and

•d

ivis

ion

num

ber

fact

s in

real

-life

si

tuat

ions

Whe

n ap

ply

ing

wit

h u

nd

erst

and

ing

lear

ners

:

use

who

le n

umb

ers

up to

mill

ions

or

•b

eyon

d in

real

-life

sit

uati

ons

use

rati

os in

real

-life

sit

uati

ons

•

use

inte

ger

s in

real

-life

sit

uati

ons

•

Learning continuums


use

the

lang

uag

e of

mat

hem

atic

s •

to c

omp

are

quan

titi

es in

real

-life

si

tuat

ions

, for

exa

mp

le, m

ore,

less

, fir

st, s

econ

d

sub

itiz

e in

real

-life

sit

uati

ons

•

use

sim

ple

frac

tion

nam

es in

real

-life

•

situ

atio

ns.

use

fast

reca

ll of

ad

dit

ion

and

•su

btr

acti

on n

umb

er fa

cts

in re

al-li

fe

situ

atio

ns

use

frac

tion

s in

real

-life

sit

uati

ons

•

use

men

tal a

nd w

ritt

en s

trat

egie

s fo

r •

add

itio

n an

d su

btr

acti

on o

f tw

o-

dig

it n

umb

ers

or b

eyon

d in

real

-life

si

tuat

ions

sele

ct a

n ap

pro

pri

ate

met

hod

for

•so

lvin

g a

pro

ble

m, f

or e

xam

ple

, m

enta

l est

imat

ion,

men

tal o

r wri

tten

st

rate

gies

, or b

y us

ing

a ca

lcul

ator

use

stra

tegi

es to

eva

luat

e th

e •

reas

onab

lene

ss o

f ans

wer

s.

use

dec

imal

frac

tion

s in

real

-life

•

situ

atio

ns

use

men

tal a

nd w

ritt

en s

trat

egie

s fo

r •

mul

tip

licat

ion

and

div

isio

n in

real

-life

si

tuat

ions

sele

ct a

n ef

ficie

nt m

etho

d fo

r •

solv

ing

a p

rob

lem

, for

exa

mp

le,

men

tal e

stim

atio

n, m

enta

l or w

ritt

en

stra

tegi

es, o

r by

usin

g a

calc

ulat

or

use

stra

tegi

es to

eva

luat

e th

e •

reas

onab

lene

ss o

f ans

wer

s

add

and

sub

trac

t fra

ctio

ns w

ith

•re

late

d d

enom

inat

ors

in re

al-li

fe

situ

atio

ns

add

and

sub

trac

t dec

imal

s in

real

-life

•

situ

atio

ns, i

nclu

din

g m

oney

esti

mat

e su

m, d

iffe

renc

e, p

rodu

ct

•an

d qu

otie

nt in

real

-life

sit

uati

ons,

in

clud

ing

frac

tion

s an

d d

ecim

als.

conv

ert i

mp

rop

er fr

acti

ons

to m

ixed

•

num

ber

s an

d vi

ce v

ersa

in re

al-li

fe

situ

atio

ns

sim

plif

y fr

acti

ons

in c

omp

utat

ion

•an

swer

s

use

frac

tion

s, d

ecim

als

and

•p

erce

ntag

es in

terc

hang

eab

ly in

real

-lif

e si

tuat

ions

sele

ct a

nd u

se a

n ap

pro

pri

ate

•se

quen

ce o

f op

erat

ions

to s

olve

wor

d p

rob

lem

s

sele

ct a

n ef

ficie

nt m

etho

d fo

r sol

ving

•

a p

rob

lem

: men

tal e

stim

atio

n, m

enta

l co

mp

utat

ion,

wri

tten

alg

orit

hms,

by

usin

g a

calc

ulat

or

use

stra

tegi

es to

eva

luat

e th

e •

reas

onab

lene

ss o

f ans

wer

s

use

men

tal a

nd w

ritt

en s

trat

egie

s fo

r •

add

ing,

sub

trac

ting

, mul

tip

lyin

g an

d d

ivid

ing

frac

tion

s an

d d

ecim

als

in

real

-life

sit

uati

ons

esti

mat

e an

d m

ake

app

roxi

mat

ions

in

•re

al-li

fe s

itua

tion

s in

volv

ing

frac

tion

s,

dec

imal

s an

d p

erce

ntag

es.

Learning continuums


No

tes

*To

cons

erve

, in

mat

hem

atic

al te

rms,

m

eans

the

amou

nt s

tays

the

sam

e re

gard

less

of t

he a

rran

gem

ent.

Lear

ners

who

hav

e b

een

enco

urag

ed

to s

elec

t the

ir ow

n ap

par

atus

and

m

etho

ds, a

nd w

ho b

ecom

e ac

cust

omed

to

dis

cuss

ing

and

ques

tion

ing

thei

r w

ork,

will

hav

e co

nfid

ence

in lo

okin

g fo

r al

tern

ativ

e ap

pro

ache

s w

hen

an in

itia

l at

tem

pt i

s un

succ

essf

ul.

Esti

mat

ion

is a

ski

ll th

at w

ill d

evel

op

wit

h ex

per

ienc

e an

d w

ill h

elp

chi

ldre

n ga

in a

“fe

el”

for n

umb

ers.

Chi

ldre

n m

ust

be

give

n th

e op

por

tuni

ty to

che

ck th

eir

esti

mat

es s

o th

at th

ey a

re a

ble

to fu

rthe

r re

fine

and

imp

rove

thei

r est

imat

ion

skill

s.

Ther

e ar

e m

any

opp

ortu

niti

es in

the

unit

s of

inqu

iry

and

duri

ng th

e sc

hool

day

for

stud

ents

to p

ract

ise

and

app

ly n

umb

er

conc

epts

aut

hent

ical

ly.

No

tes

**M

odel

ling

invo

lves

usi

ng c

oncr

ete

mat

eria

ls to

rep

rese

nt n

umb

ers

or

num

ber

op

erat

ions

, for

exa

mp

le, t

he u

se

of p

atte

rn b

lock

s or

frac

tion

pie

ces

to

rep

rese

nt fr

acti

ons

and

the

use

of b

ase

10

blo

cks

to re

pre

sent

num

ber

op

erat

ions

.

Stud

ents

nee

d to

use

num

ber

s in

m

any

situ

atio

ns in

ord

er to

ap

ply

thei

r un

der

stan

din

g to

new

sit

uati

ons.

In

add

itio

n to

the

unit

s of

inqu

iry,

chi

ldre

n’s

liter

atur

e al

so p

rovi

des

rich

op

por

tuni

ties

fo

r dev

elop

ing

num

ber

con

cep

ts.

To b

e us

eful

, ad

dit

ion

and

sub

trac

tion

fa

cts

need

to b

e re

calle

d au

tom

atic

ally

. Re

sear

ch c

lear

ly in

dic

ates

that

ther

e ar

e m

ore

effe

ctiv

e w

ays

to d

o th

is th

an “d

rill

and

pra

ctic

e”. A

bov

e al

l, it

hel

ps

to h

ave

stra

tegi

es fo

r wor

king

them

out

. Cou

ntin

g on

, usi

ng d

oub

les

and

usin

g 10

s ar

e g

ood

stra

tegi

es, a

ltho

ugh

lear

ners

freq

uent

ly

inve

nt m

etho

ds th

at w

ork

equa

lly w

ell f

or

them

selv

es.

Dif

ficul

ties

wit

h fr

acti

ons

can

aris

e w

hen

frac

tion

al n

otat

ion

is in

trod

uced

bef

ore

stud

ents

hav

e fu

lly c

onst

ruct

ed m

eani

ng

abou

t fra

ctio

n co

ncep

ts.

No

tes

Mod

ellin

g us

ing

man

ipul

ativ

es p

rovi

des

a

valu

able

sca

ffol

d fo

r con

stru

ctin

g m

eani

ng a

bou

t mat

hem

atic

al

conc

epts

. The

re s

houl

d b

e re

gula

r op

por

tuni

ties

for l

earn

ers

to w

ork

wit

h a

rang

e of

man

ipul

ativ

es a

nd to

d

iscu

ss a

nd n

egot

iate

thei

r dev

elop

ing

und

erst

and

ings

wit

h ot

hers

.

***E

xam

ple

s of

rela

ted

den

omin

ator

s in

clud

e ha

lves

, qua

rter

s (fo

urth

s) a

nd

eigh

ths.

The

se c

an b

e m

odel

led

easi

ly b

y fo

ldin

g st

rip

s or

squ

ares

of p

aper

.

The

inte

rpre

tati

on a

nd m

eani

ng o

f re

mai

nder

s ca

n ca

use

dif

ficul

ty fo

r so

me

lear

ners

. Thi

s is

esp

ecia

lly tr

ue if

ca

lcul

ator

s ar

e b

eing

use

d. F

or e

xam

ple

, 67

÷ 4

= 1

6.75

. Thi

s ca

n al

so b

e sh

own

as 1

6¾ o

r 16

r3. L

earn

ers

need

pra

ctic

e in

pro

duci

ng a

pp

rop

riat

e an

swer

s w

hen

usin

g re

mai

nder

s. F

or e

xam

ple

, for

a

scho

ol tr

ip w

ith

25 s

tud

ents

, onl

y b

uses

th

at c

arry

20

stud

ents

are

ava

ilab

le. A

re

mai

nder

cou

ld n

ot b

e le

ft b

ehin

d, s

o an

othe

r bus

wou

ld b

e re

quire

d!

Cal

cula

tor 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

flec

ting

on

and

reco

rdin

g th

eir

find

ings

in m

athe

mat

ics

lear

ning

logs

, st

uden

ts b

egin

to n

otic

e p

atte

rns

in th

e nu

mb

ers

that

will

furt

her d

evel

op th

eir

und

erst

and

ing.

No

tes

It is

not

pra

ctic

al to

con

tinu

e to

dev

elop

an

d us

e b

ase

10 m

ater

ials

bey

ond

1,00

0.

Lear

ners

sho

uld

have

litt

le d

iffic

ulty

in

exte

ndin

g th

e p

lace

val

ue s

yste

m o

nce

they

hav

e un

der

stoo

d th

e gr

oup

ing

pat

tern

up

to 1

,000

. The

re a

re a

num

ber

of

web

site

s w

here

vir

tual

man

ipul

ativ

es

can

be

utili

zed

for w

orki

ng w

ith

larg

er

num

ber

s.

Esti

mat

ion

pla

ys a

key

role

in c

heck

ing

the

feas

ibili

ty o

f ans

wer

s. T

he m

etho

d of

mul

tip

lyin

g nu

mb

ers

and

igno

ring

th

e d

ecim

al p

oint

, the

n ad

just

ing

the

answ

er b

y co

unti

ng d

ecim

al p

lace

s, d

oes

not g

ive

the

lear

ner a

n un

der

stan

din

g of

w

hy it

is d

one.

Ap

plic

atio

n of

pla

ce v

alue

kn

owle

dg

e m

ust p

rece

de

this

ap

plic

atio

n of

pat

tern

.

Mea

sure

men

t is

an e

xcel

lent

way

of

exp

lori

ng th

e us

e of

frac

tion

s an

d d

ecim

als

and

thei

r int

erch

ang

e.

Stud

ents

sho

uld

be

give

n m

any

opp

ortu

niti

es to

dis

cove

r the

link

b

etw

een

frac

tion

s an

d d

ivis

ion.

A th

orou

gh u

nder

stan

din

g of

m

ulti

plic

atio

n, fa

ctor

s an

d la

rge

num

ber

s is

requ

ired

bef

ore

wor

king

wit

h ex

pon

ents

.


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.

Documents

Mathematics scope and sequence - Älmhults · PDF fileMathematics scope and sequence 1 Introduction to the PYP mathematics scope and sequence The information in this scope and sequence