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3The National Strategies | SecondarySecondary mathematics subject leader development materials: Spring 2010

TheFrameworkforsec

ondarymathematics:overviewan

dlearningobjectives

Overviewofstrands

Strands

Sub-strands

Strands

Sub-strands

1Mathe

maticalprocessesandapplications

3Algebra

1.1

Representing

3.1

Equations,formulae,expressionsandidentities

1.2

Analysingusemathematicalreasoning

3.2

Sequences,functionsandgraphs

1.3

Analysinguseapp

ropriatemathematicalprocedures

4Geometryandmeasures

1.4

Interpretingandevaluating

4.1

Geometricalreasoning

1.5

Communicatingandreflecting

4.2

Transformationsandcoordinates

2Numb

er

4.3

Constr

uctionandloci

2.1

Placevalue,orderin

gandrounding

4.4

Measu

resandmensuration

2.2

Integers,powersan

droots

5Statistics

2.3

Fractions,decimals,percentages,ratioandproportion

5.1

Specifyingaproblem,planningandcollectingdata

2.4

Numberoperations

5.2

Proces

singandrepresentingdata

2.5

Mentalcalculationmethods

5.3

Interpretinganddiscussingresults

2.6

Writtencalculationmethods

5.4

Probability

2.7

Calculatormethods

2.8

Checkingresults

Crown copyright 2009 01061-2009DOM-EN


2/42


3/42

5

Learn

ingobjectives

1Mathematicalprocessesanda

pplications

Solvepro

blems,exploreandinvestigate

inarangeofcontexts

Increasethechallengeandbuildprogressionacrossthekeystage,andfor

groupsofpupilsby:

tincrea

singthecomplexityoftheapplication,e.g.non-routine,multi-stepproblems,extendedenquiries

treduc

ingthefamiliarityofthecontext,e.g.newcontextsinmathematics,contextsdrawnfromothersu

bjects,otheraspectsofpupilsliv

es

The National Strategies | SecondarySecondary mathematics subject leader development materials: Spring 2010


tincrea

singthetechnicaldemandofth

emathematicsrequired,e.g.mo

readvancedconcepts,moredifficultprocedures

tincrea

singthedegreeofindependenc

eandautonomyinproblem-solv

ingandinvestigation

Representing

1.1 id

entifythenecessary

informat

ionto

understa

ndorsimplify

acontextorproblem;

represen

tproblems,

makingcorrectuse

ofsymbols,words,

diagrams,tablesand

graphs;u

seappropriate

proceduresandtools,

includingICT

Year7

Year8

Year9

Year10

Year11

Extension

identifythe

mathematicalfeatures

ofacontextor

problem;tryout

and

comparemathem

atical

representations;

select

appropriateproc

edures

andtools,includ

ingICT

breakdownsubstantial

taskstomakethem

moremanageable;

representproblems

andsynthesise

informationin

algebraic,geometrical

orgraphicalform;

movefromoneform

toanothertogaina

differentperspective

ontheproblem

compareandevaluate

representations;

explainthefeatures

selectedandjustify

thechoiceof

representationin

relationtothecontext

chooseandcombine

representationsfroma

rang

eofperspectives;

intro

duceandusea

rang

eofmathematical

tech

niques,themost

effic

ientforanalysis

and

mosteffectivefor

com

munication

systematicallymodel

contextsorproblems

throughprecise

andconsiste

ntuse

ofsymbolsa

nd

representations,and

sustainthisthroughout

thework


4/42

6

Ana

lysingusemathematic

alreasoning

1.2 Y

ear7

Year8

Year9

Year10

Year11

Extension


01061-2009DOM-EN Crown copyright 2009

classifya

ndvisualise

propertiesand

patterns;generalise

insimple

casesby

working

logically;draw

simpleconclusions

andexplainreasoning;

understa

ndthe

significanceofa

counter-example;

takeacco

untof

feedbackandlearn

frommis

takes

visualiseand

manipulatedyna

mic

images;conjectu

re

andgeneralise;m

ove

betweenthegen

eral

andtheparticula

rto

testthelogicofan

argument;identify

exceptionalcase

sor

counter-examples;

makeconnectionswith

relatedcontexts

useconnectionswith

relatedcontextsto

improvetheanalysisof

asituationorproblem;

posequestionsand

makeconvincing

argumentstojustify

generalisationsor

solutions;recognisethe

impactofconstraintsor

assumptions

identifyarange

ofstrategiesand

appreciatethatmore

thanoneapproach

maybenecessary;

exploretheeffects

ofvaryingvaluesand

lookforinvarianceand

covarianceinmodels

andrepresentations;

examineandrefine

arguments,conclusions

andgeneralisations;

producesimpleproofs

makeprogressby

exploringmathematical

tasks,developing

andfollowing

alter

nativeapproaches;

exam

ineandextend

generalisations;support

assumptionsbyclear

argu

mentandfollow

throughasustained

chainofreasoning,

inclu

dingproof

presentrigorousand

sustainedarguments;

reasoninductively,

deduceand

prove;

explainandjustify

assumptionsand

constraints

Ana

lysinguseappropriate

mathematicalprocedures

1.3Withinthe

appropriaterangeandcontent:

makeaccu

ratemathematicaldiagrams,gra

phsandconstructionsonpaperandonscreen;calculateaccurate

ly,selectingmentalmethodsorc

alculating

devicesas

appropriate;manipulatenumbers,algebraicexpressionsandequ

ations,andapplyroutinealgorith

ms;useaccuratenotation,includ

ingcorrect

syntaxwh

enusingICT;recordmethods,so

lutionsandconclusions;estimate

,approximateandcheckworking


5/42


6/42


7/42

9

2Num

ber

Plac

evalue,orderingandrounding

2.1



Year7

Year8

Year9

Year10

Year11

Extension

understa

ndanduse

decimalnotationand

placevalue;multiply

anddivid

eintegers

anddecimalsby10,

100,1000,andexplain

theeffec

t

compare

andorder

decimals

indifferent

contexts

;knowthat

whencomparing

measure

mentsthe

unitsmu

stbethesame

roundpo

sitivewhole

numberstothenearest

10,100o

r1000,and

decimals

tothenearest

wholenu

mberorone

decimalplace

readandwritepositive

integerpowerso

f10;

multiplyanddivide

integersanddec

imals

by0.1and0.01

orderdecimals

roundpositivenumbers

toanygivenpow

er

of10;rounddecimals

tothenearestwhole

numberortooneor

twodecimalplac

es

extendknowledge

ofintegerpowers

of10;recognisethe

equivalenceof0.1,101

and101;multiplyand

dividebyanyinteger

powerof10

useroundingtomake

estimatesandtogive

solutionstoproblems

toanappropriate

degreeofaccuracy

convertbetween

ordinaryand

standardindexform

representations,using

significantfiguresas

appropriate;justifythe

representationused

andchoiceofaccuracy

inrelationtothe

problemandaudience

forthesolution

engagein

mathematicaltasks

whereusingnumbers

instandardform

isessentialtothe

calculationsinvolved;

critic

allyexaminethe

effectofnumerical

representations

ontheaccuracyof

thesolution,e.g.

understandhowerrors

canbecompounded

inca

lculations

communicatethe

solutiontoa

problem,

explainingthe

limitationso

faccuracy,

usingupper

and

lowerbound

s


8/42

10

Inte

gers,powersandroots

2.2 Ye

ar7

Year8

Year9

Year10

Year11

understa

ndnegative

numbers

aspositions

onanum

berline;

order,ad

dandsubtract

integers

incontext

recogniseanduse

multiples,factors,

primes(lessthan100),

common

factors,

highestc

ommon

factorsandlowest

common

multiplesin

simplecases;usesimple

testsofd

ivisibility

recognisethefirstfew

triangula

rnumbers;

recognisethesquares

ofnumberstoat

least12

12andthe

correspo

ndingroots

add,subtract,multiply

anddivideintegers

usemultiples,factors,

commonfactors,

highestcommon

factors,lowest

commonmultiplesand

primes;findtheprime

factordecomposition

ofanumber,e.g.

8000=26

53

usesquares,positive

andnegativesqu

are

roots,cubesand

cuberoots,andindex

notationforsma

ll

positiveintegerpowers

usetheprimefactor

decompositionof

anumber

useICTtoestimate

squarerootsand

cuberoots

useindexnotationfor

integerpowers;know

andusetheindexlaws

formultiplicationand

divisionofpositive

integerpowers

examineandextend

theindexlawsto

establishthemeaning

ofnegative,fractional

andzeropowers,

includinguseof

surdnotation

exam

ineandextend

theindexlawsto

establishthemeaning

ofin

verseoperationsin

relat

iontoindices,i.e.

theinverseoperation

ofra

isingapositive

num

bertopowernis

raisingtheresultofthis

operationtopower1n

Extension

solveaprob

lem

usingrationaland

irrationalnumbers,

includingsurds




9/42

11

Frac

tions,decimals,percent

ages,ratioandproportion

2.3 Ye

ar7

Year8

Year9

Year10

Year11

Extension



expressasmallerwhole

numberasafractionof

alargero

ne;simplify

fractions

bycancelling

allcomm

onfactors

andiden

tifyequivalent

fractions

;convert

terminat

ingdecimals

tofractio

ns,e.g.

2

3

=

0.23

10

0;use

diagramstocompare

twoorm

oresimple

fractions

addand

subtract

simplefr

actionsand

thosewithcommon

denomin

ators;

calculate

simple

fractions

ofquantities

andmea

surements

(whole-n

umber

answers);multiplya

fractionbyaninteger

recognisethata

recurringdecimalisa

fraction;usedivisionto

convertafractiontoa

decimal;orderfractions

bywritingthem

witha

commondenom

inator

orbyconverting

them

todecimals

understandthe

equivalenceofsimple

algebraicfractions;

knowthatarecurring

decimalisanexact

fraction

explainthepatterns

foundinrecurring

decimals;justify

whydecimalsrecur

orterminateby

consideringfactorsof

thedenominator

explorethehistorical

andculturalrootsof

thenumbersystem

and

usealgebrato

justifyandprovesome

ofitsfeatures,e.g.that

allre

curringdecimals

canbeexpressedas

afraction

showinsightintothe

infinitedens

ityofthe

numberline;make

senseofthe

proofthat

2isirration

al

addandsubtract

fractionsbywriting

themwithacom

mon

denominator;calculate

fractionsofquan

tities

(fractionanswers);

multiplyanddividean

integerbyafraction

useefficientmethodsto


anddividefractions,

interpretingdivisionas

amultiplicativeinverse;

cancelcommonfactors

beforemultiplying

ordividing

understandandapply

efficientmethodsto


anddividefractions,

interpretingreciprocals

asmultiplicative

inverses


10/42

12

Frac

tions,decimals,percent

ages,ratioandproportion(continued)

2.3 Ye

ar7

Year8

understa

ndpercentage

asthenumberofparts

per100;

calculate

simplepercentages

andusepercentages

tocomparesimple

proportions

recognisethe

equivalenceof

percenta

ges,fractions

anddecimals

understa

ndthe

relations

hipbetween

ratioand

proportion;

usedirec

tproportion

insimple

contexts;use

rationotation,simplify

ratiosanddividea

quantity

intotwo

partsinagivenratio;

solvesim

pleproblems

involving

ratioand

proportionusing

informal

strategies

interpretpercentage

astheoperatorso

manyhundredth

s

ofandexpresso

ne

givennumberas

a

percentageofan

other;

calculatepercentages

andfindtheoutcome

ofagivenpercen

tage

increaseordecre

ase

usetheequivalence

offractions,decimals

andpercentages

to

compareproportions

applyunderstanding

oftherelationship

betweenratioan

d

proportion;simp

lify

ratios,including

those

expressedindiff

erent

units,recognisinglinks

withfractionnotation;

divideaquantity

into

twoormorepartsin

agivenratio;use

the

unitarymethodto

solvesimpleproblems

involvingratioand

directproportion

Year9

Year10

Year11

Extension

recognisewhen

fractionsor

percentagesare

neededtocompare

proportions;solve

problemsinvolving

percentagechanges

useproportional

reasoningtosolve

problems,choosing

thecorrectnumbers

totakeas100%,oras

awhole;comparetwo

ratios;interpretand

useratioinarange

ofcontexts

identifywhena

probleminnumber,

algebra,geometry

orstatisticsinvolves

proportionality;use

multiplicativemethods

fluentlyinthesolution,

includinginverse

calculations,e.g.

withpercentages

mod

elrealcontexts

wherequantitiesvary

indirectproportion,

inclu

dingrepeated

prop

ortionalchange,

e.g.growth/decay;use

algebraicmethods

whereappropriateand

cons

iderlimitationsof

themodel

(asin

2.4

)

understandanduse

directandin

verse

proportion;solve

problemsinvolving

inverseprop

ortion

(includingy?1

/x2)

(asin2

.4)




11/42

13

Num

beroperations

2.4 Ye

ar7

Year8

Year9

Year10

Year11

Extension



understa

ndandusethe

understandand

usethe

understandtheeffects

recogniseanduse

mod

elrealcontexts

understandanduse

rulesofa

rithmeticand

rulesofarithmet

icand

ofmultiplyingand

reciprocalsasa

wherequantitiesvary

directandin

verse

inverseo

perationsin

inverseoperationsin

dividingbynumbers

multiplicativeinverse

indirectproportion,

proportion;solve

thecontextofpositive

thecontextofintegers

between0and1;

incontextssuchas

inclu

dingrepeated

problemsinvolving

integers

anddecimals

andfractions

consolidateuseofthe

enlargement;explore

prop

ortionalchange,

inverseprop

ortion

rulesofarithmeticand

thebehaviourofthe

growth/decay;use

e.g.

(includingy?

1/x2)

inverseoperations

reciprocalfunction

algebraicmethods

(asin2

.3)

(y=

/x)forlargeand

whereappropriateand

smallvaluesofx

cons

iderlimitationsof

themodel

(asin

2.3

)

usetheo

rderof

usetheorderof

understandtheorder

operations,including

operations,inclu

ding

ofprecedenceof

brackets

brackets,withm

ore

operations,including

complexcalculations

powers


12/42

14

Men

talcalculationmethods

2.5 Ye

ar7

Year8

Year9

Year10

Year11

Extension



recallnumber

facts,inc

luding

positiveinteger

complem

entsto100

andmultiplicationfacts

to1010,andquickly

deriveas

sociated

divisionfacts

strengthenandextend

mentalm

ethodsof

calculationtoinclude

decimals

,fractions

andperc

entages,

accompa

niedwhere

appropriatebysuitable

jottings;

solvesimple

problem

smentally

makeandjustify

estimate

sand

approxim

ationsof

calculations

recallequivalent

fractions,decimals

andpercentages

;

useknownfacts

to

deriveunknown

facts,

includingproducts

involvingnumbe

rs

suchas0.7and6

,and

and8

0.03

strengthenandextend

mentalmethods

of

calculation,work

ing

withdecimals,

fractions,percen

tages,

squaresandsquare

roots,cubesand

cube

roots;solveprob

lems

mentally

useknownfactsto

deriveunknown

facts;extendmental

methodsofcalculation,

workingwithdecimals,

fractions,percentages,

factors,powersand

roots;solveproblems

mentally

selectmentalor

writtenstrategies

andcalculating

devicesappropriate

tothestageofthe

problem;calculate

accuratelywith

reciprocals,powers,

trigonometrical

functionsandnumbers

instandardform

(asin2

.6and2

.7)

selectandjustify

anappropriateand

effic

ientcombination

ofm

ethodsof

calculation,i.e.

men

tal,written,ICT

orca

lculatortosolve

prob

lems

(asin

2.6

)

appreciatew

henresults

ofcalculationscanbe

moreelegan

tlyand

exactlycommunicated

usingsurdsand

,rationalisinga

denominato

rwhere

appropriate,e.g.

atrigonome

trical

solution

(asin2

.6)

makeandjustify

makeandjustify

examineandrefine

estimatesand

estimatesand

estimatesand

approximationsof

approximationsof

approximationsof

calculations

calculations

calculationsinvolving

rounding


13/42

15

Writtencalculationmethods

2.6 Ye

ar7

Year8

Year9

Year10

Year11

Extension

useefficientwritten

methods

toadd

andsubt

ractwhole

numbers

anddecimals

withuptotwoplaces

multiply

anddivide

three-digitbytwo-

digitwholenumbers;

extendtomultiplying

anddivid

ingdecimals

withone

ortwoplaces

bysingle

-digitwhole

numbers

useefficientwrit

ten

methodstoaddand

subtractintegersand

decimalsofanysize,

includingnumbers

withdifferingnu

mbers

ofdecimalplaces



useefficientwrit

ten

methodsfor

multiplicationan

d

divisionofintegersand

decimals,includingby

decimalssuchas

0.6or

0.06;understand

where

topositionthed

ecimal

pointbyconside

ring

equivalentcalculations

useefficientwritten

methodstoaddand

subtractintegersand

decimalsofanysize;

multiplybydecimals;

dividebydecimals

bytransformingto

divisionbyaninteger

selectmentalor

writtenstrategies

andcalculating

devicesappropriate

tothestageofthe

problem;calculate

accuratelywith

reciprocals,powers,

trigonometrical

functionsandnumbers

instandardform

(asin2

.5and2

.7)

selectandjustify

anappropriateand

effic

ientcombination

ofm

ethodsof

calculation,i.e.

men

tal,written,ICT

orca

lculatortosolve

prob

lems

(asin

2.5

)

appreciatew

henresults

ofcalculationscanbe

moreelegan

tlyand

exactlycommunicated

usingsurdsand

,rationalisinga

denominato

rwhere

appropriate,e.g.

atrigonome

trical

solution

(asin2

.5)


14/42

16

Calc

ulatormethods

2.7 Ye

ar7

Year8

Year9

Year10

Year11

Extension

carryoutcalculations

withmorethanone

stepusin

gbracketsand

themem

ory;usethe

squarero

otandsign

changek

eys

enternumbersand

interpret

thedisplay

indiffere

ntcontexts

(decimals,percentages,

money,m

etric

measures)

carryoutmored

ifficult

calculationseffectively

andefficientlyusing

thefunctionkeysfor

signchange,pow

ers,

rootsandfractio

ns;

usebracketsand

thememory

useacalculator

efficientlyand

appropriatelyto

performcomplex

calculationswith

numbersofanysize,

knowingnottoround

duringintermediate

stepsofacalculation;

usetheconstant,

andsignchangekeys;

usethefunctionkeys

forpowers,rootsand

fractions;usebrackets

andthememory

selectmentalor

writtenstrategies

andcalculating

devicesappropriate

tothestageofthe

problem;calculate

accuratelywith

reciprocals,powers,

trigonometrical

functionsandnumbers

instandardform

(asin2

.5and2

.6)

critic

allyexamine

alter

nativemethods,

comparestrategiesfor:

tc

alculating

(including

calculatingdevices)

tc

hecking

reco

gnisethe

limit

ationsofsome

approaches

(asin

2.8

)

reflectonasolutionto

aproblemcommenting

constructive

lyonthe

choiceofcalculating

strategies

(asin2

.8)

enternumbersand

interpretthedisplay

indifferentcontexts

(extendtonegat

ive

numbers,fractions,

time)




15/42

17

Che

ckingresults

2.8 Ye

ar7

Year8

Year9

Year10

Year11

Extension

checkresultsby

consideringwhether

theyare

oftheright

orderofmagnitude

andbyw

orking

problem

sbackwards

selectfromaran

ge

checkresultsusing

identifyarangeof

ofcheckingmethods,

appropriatemethods

checkingstrategiesand

includingestima

ting

appreciatethatmore

incontextandusing

thanonewaymay

inverseoperations

benecessaryinthe

contextoftheproblem

critic

allyexamine

alter

nativemethods,

com

parestrategiesfor:

tc

alculating

(including

calculatingdevices)

tc

hecking

reco

gnisethe

limit

ationsofsome

approaches

(asin

2.7

)

reflectonasolutionto

aproblemcommenting

constructive

lyonthe

choiceofchecking

strategies

(asin2

.7)




16/42


17/42

19

3Algebra

Equations,formulae,expressionsandidentities

3.1



Year7

Year8

Year9

Year10

Year11

Extension

uselettersymbolsto

recognisethatle

tter

distinguishthe

presentconvincing

exam

ineandrefine

usesymbols

and

represen

tunknown

symbolsplaydifferent

differentrolesplayed

algebraicargumentsto

algebraicarguments

representati

ons

numbers

orvariables;

rolesinequation

s,

bylettersymbolsin

justifygeneralisations

pres

entedtoexplain

consistently

topresent

knowthemeanings

formulaeandfunctions;

equations,identities,

orsolutions;relate

geometricaland

aformalpro

of,e.g.

ofthewordsterm,

knowthemeanings

formulaeandfunctions

argumentstothe

num

ericalproperties;

derivingthe

formula

expressio

nandequation

ofthewordsform

ula

structureofthecontext

chooseandcombine

forsolvingq

uadratic

andfunction

orproblem;produce

representationsto

equations

simpleproofs

pres

entaconvincing

proo

f

understa

ndthat

understandthat

useindexnotationfor

usealgebraic

appreciatethe

algebraicoperations

algebraicoperations,

integerpowersand

representationto

generalityoftheforms

followth

erulesof

includingtheuseof

simpleinstancesof

synthesiseknownrules

a+b

=candab=c,

arithmet

ic

brackets,followthe

theindexlaws

ofarithmetic,including

whe

reeachtermcan

rulesofarithmet

ic;use

thecommutativeand

itselfbeanexpression;

indexnotationforsmall

distributivelaws;justify

usethisinsightinto

positiveintegerpowers

thesegeneralisations,

structuretodevelop

usingspatial

e.g.

fluencyintransforming

representations;use

morecomplex

algebraicargumentto

equations

generalisetheindex

lawsformultiplication

anddivisiontoinclude

zero,negativeand

fractionalpowers


18/42

20

Equations,formulae,expressionsandidentities(con

tinued)

3.1 Ye

ar7

Year8

Year9

Year10

Year11

Extension

simplifylinearalgebraic

expressionsby

collectingliketerms;

multiply

asingleterm

overabr

acket(integer

coefficients)

simplifyortransform

linearexpression

sby

collectingliketerms;

multiplyasingle

term

overabracket

simplifyortransform

algebraicexpressions

bytakingoutsingle-

termcommonfactors;

addsimplealgebraic

fractions

developfluencyin

transforminglinear

expressions;expand

theproductoftwo

linearexpressions

oftheformxn

andfactorisesimple

quadraticexpressions;

establishidentitiessuch

asthedifferenceof

twosquares;compare

andevaluatedifferent

representationsofthe

samecontext;identify

equivalentexpressions

andconfirmby

transformation

expa

ndandfactorise

quadraticexpressions;

simp

lifyortransform

alge

braicfractions,

byfactorisingand

e.g.canc

ellingcommon

factors;compareand

evaluatedifferent

representationsofthe

samecontext;identify

equivalentexpressions

and

confirmthisby

transformation




19/42

21


tinued)

3.1 Ye

ar7

Year8

Year9

Year10

Year11

Extension

construc

tand

solvesim

plelinear

equation

swithinteger

coefficients(unknown

ononesideonly)using

anappro

priatemethod

inve

rseoperations)

(e.g.

constructandso

lve

linearequations

with

integercoefficients

(unknownoneitheror

bothsides,witho

utand

withbrackets)us

ing

appropriatemethods

inverseoperations,

(e.g.

transformingboth

sidesinsamewa

y)

usegraphsands

et

upequationstosolve

simpleproblems

involvingdirect

proportion

constructandsolve

linearequationswith

integercoefficients

(withandwithout

brackets,negative

signsanywhereinthe

equation,positiveor

negativesolution)

usealgebraicmethods

tosolveproblems

involvingdirect

proportion;relate

algebraicsolutions

tographsofthe

equations;useICT

asappropriate

constructlinear

equationsandsimple

linearinequalities(one

variable)torepresent

real-lifesituations

ormathematical

problems;solve

linearequations

andinequalities,

representingthe

solutioninthecontext

oftheproblem

cons

tructsimple

quadraticequations

tore

presentreal-

lifes

ituationsor

mathematicalproblems

and

solvethem

usingfactorisation,

grap

hicalortrialand

improvementmethods;

justifythenumber

ofso

lutionsusing

alge

braicorgraphical

argu

mentsandselect

appropriatesolutions,

interpretingtheir

accu

racy

representreal-

lifesituation

sor

mathematicalproblems

involving:

tmore

co

mplex

quadrat

ic

equations,

choosin

gan

appropriate

method

of

solution

including

completingthe

squareanduseof

theform

ula

tdirec

torinverse

proport

ion,

includin

g

y?

x2,

y?1/x

2

relatealgebraic

solutionsto

graphical

representati

onofthe

functions




20/42

22


tinued)

3.1 Ye

ar7

Year8

Year9

Year10

Year11

Extension

usesystematictrialand

improvementmethods

andICTtoolstofind

approximatesolutions

toequationssuchas

x2+

x=20

explorewaysof

constructingmodels

ofreal-lifesituations

bydrawinggraphs

andconstructing

algebraicequationsand

inequalities

(Seeobjectiveabove

(Seeobjectiveabove

(Seeobjectiveabove

forprogression)

forprogression)

forprogression)

constructapairof

simultaneouslinear

equationstorepresent

real-lifesituations

ormathematical

problems;examine

andcompare

algebraicmethodsof

solution;usegraphical

representationto

explainwhythe

intersectionoftwo

linesgivesthecommon

solutionandwhysome

caseshavenocommon

solutionandothers

haveaninfinitenumber

selectandjustify

optimummethods

forsolvingapair

ofsimultaneous

linea

requationsina

varie

tyofcontexts;

cons

tructseveral

linea

rinequalities

inon

eandtwo

varia

blestorepresent

real-

lifesituations

orm

athematical

prob

lems;solve

theinequalities

grap

hically,identifying

and

interpretingthe

solutionsetinthe

cont

extoftheproblem

solvemorec

omplex

pairsofsimu

ltaneous

equationsgenerated

fromreal-lifecontexts

orgeometrical

investigations,

includingpa

irswhere

oneislinear

andthe

otherisquadraticorof

theformx2 +

y2=

r2




21/42

23


tinued)

3.1 Ye

ar7

Year8

Year9

Year10

Year11

Extension

usesimp

leformulae

frommathematics

andothe

rsubjects;

substitutepositive

integers

intolinear

expressionsand

formulae

and,insimple

cases,de

riveaformula

useformulaefrom

mathematicsandother

subjects;substitute

integersintosim

ple

formulae,including

examplesthatle

ad

toanequationto

solve;substitute

positiveintegers

into

expressionsinvo

lving

smallpowers,e.g

.

3x2+

4or2x3;de

rive

simpleformulae

useformulaefrom

mathematicsand

othersubjects;

substitutenumbers

intoexpressionsand

formulae;derivea

formulaand,insimple

cases,changeits

subject

deriveformulae,e.g.

inthecontextof

mensuration;interpret

arangeofformulae

drawnfromreal-life

contextsandother

subjects,relating

thevariablestothe

contextanddescribing

theirbehaviour;

solveproblemsby

manipulatingformulae

deriveanduse

form

ulaethatinvolve

morevariables

orm

orecomplex

alge

braicexpressions;

man

ipulateformulae

inor

dertoreacha

solution,showinsight

into

themathematical

conn

ections,e.g.using

thecontextandthe

form

ulaetoexplainthe

prop

ortionaleffectof

vary

ingvalues




22/42

24

Sequences,functionsandgraphs

3.2 Ye

ar7

Year8

Year9

Year10

Year11

Extension

describe

integer

sequences;generate

termsof

asimple

sequence,givena

rule(e.g.

findinga

termfromtheprevious

term,findingaterm

givenits

positionin

thesequ

ence)

generatetermso

f

alinearsequence

usingterm-to-term

andposition-to-term

rules,onpapera

nd

usingaspreadsh

eetor

graphicscalculator

generatetermsofa

sequenceusingterm

to-termandposition

to-termrules,onpaper

andusingICT

generate

sequences

frompat

ternsor

practical

contextsand

describe

thegeneral

termins

implecases

develop,compareand

evaluatealgebraicand

spatialrepresentations

ofsituationsthat

generatesequences;

interpret,deduceand

justifygeneralisations

forthenthtermof

linearandquadratic

sequences,including

thepropertiesof

squareandtriangular

numbers



uselinearexpressions

todescribethenthterm

ofasimplearithm

etic

sequence,justifying

itsformbyreferringto

theactivityorpractical

contextfromwhichit

wasgenerated

generatesequences

frompracticalcontexts

andwriteandjustifyan

expressiontodescribe

thenthtermofan

arithmeticsequence


23/42

25

Sequences,functionsandgraphs(continued)

3.2 Ye

ar7

Year8

Year9

Year10

Year11

Extension

expresssimple

expresssimple

findtheinverseofa

comparegraphical,

functionsinwords,

functionsalgebraically

linearfunction

algebraicand

thenusin

gsymbols;

andrepresentth

em

geometrical

represen

tthemin

inmappingsoro

na

representations,

mapping

s

spreadsheet

includingmapping

diagrams,toexplain

theeffectof:

trotat

ingtheline

y=mx+cthrough

90aboutany

point

trefle

ctingtheline

y=mx+cinthe

liney=x

derivepropertiesof

perpendicularlinesand

oftheinversefunction

generate

coordinate

generatepointsinall

generatepointsand

exploregraphsof

expl

oreconnections

exploregrap

hsof

pairstha

tsatisfya

fourquadrantsa

ndplot

plotgraphsoflinear

functionsoftheform

betw

eentheformof

exponential

and

simplelinearrule;plot

thegraphsoflinear

functions,whereyis

y=xn(

naninteger)

theequationandthe

trigonometrical

andrecognisetheir

resultinggraphsof

functionsan

drecognise

thegraphsofsimple

functions,where

yis

givenimplicitlyinterms

linearfunctions,where

givenexplicitlyinterms

ofx(e.g.a

y+bx=0,

characteristicshapes;

quadraticandcubic

theircharact

eristic

yisgiven

explicitly

ofx,onpaperan

d

y+bx+c=0),onpaper

varythevaluesofa,b

func

tionssuchas:

shapes;applytothe

interms

ofx,on

usingICT;recogn

ise

andusingICT;findthe

andcinfunctionssuch

graphy=f(x)the

ty=(x+2)(x5)

paperan

dusingICT;

thatequationsof

gradientoflinesgiven

asy=ax2+

c,

transformations

ty=

(x2)(x2

+7x+12)

recognisestraight-line

theformy=mx+c

byequationsofthe

y=ax3+

c,

y=

f(x)+a,y

=af(x),

graphsp

aralleltothe

correspondtostraight-

formy=mx+c,given

y=(x+b)2usinga

y=

f(x+a),y

=f(ax)for

ty=x2

2x+1

x-axisor

y-axis

linegraphs

valuesformandc

graphplottertoexplain

linear,quadr

atic,sine

ty=x3+

3

howthistransforms

andcosinefunctions;

thegraph

useagraphplotterto

inclu

defeaturessuch

explaintheeffectof

asro

otsoftheequation,

transformationsonthe

interceptsandturning

graphandgeneralise

poin

ts

tootherfunctions




24/42

26


3.2 Ye

ar7

Year8

Year9

Year10

Year11

Extension

plotand

interpretthe

constructlinear

constructfunctions

sketchandinterpret

applyknowledgeof

setupamat

hematical

graphso

fsimplelinear

functionsarising

from

arisingfromreal-life

graphsthatmodelreal-

mathematicalfunctions

modelofareal-life

functionsarisingfrom

real-lifeproblem

s

problemsandplottheir

lifesituations,including

toproblemsinvolving:

contextorp

roblem,

real-lifes

ituations,e.g.

andplottheir

correspondinggraphs;

thosegenerated

identifyingt

he

to

ptimisation,

conversiongraphs

correspondingg

raphs;

interpretgraphsarising

fromothersubjects

variablesandtheir

usingnumerical,

discussandinter

pret

fromrealsituations,e.g.

suchasscience;

functionalre

lationship;

algebraicand

graphsarisingfrom

timeseriesgraphs

usemathematical

usegraphsa

nd

graphical,

realsituations,e.g.

argumenttojustify

sketchestoexplain

techniques,

distancetimegraphs

featuresoftheirshapes

thebehaviourofthe

includingmaxima

variablesandtoexplain

andminima

orjustifythe

effectof

tu

singICTtofit

assumptionsin

acurvetodata

themodel

fromarealcontext

suchasascience

experiment

tre

peated

proportional

change,e.g.

compoundinterest

useICTtoexplorethe

graphicalrepresentation

ofalgebraicequations

andtointerprethow

propertiesofthegraph

arerelatedtofeatures

oftheequation,

paralleland

e.g.

perpendicularlines




25/42

27


3.2 Ye

ar7

Year8

Year9

Year10

Year11

Extension



interpretthemeaning

ofvariouspointsand

sectionsofstraight-

linegraphs,including

interceptsand

intersections,e.g.s

olving

simultaneouslinear

equations


26/42


27/42

29

4Geo

metryandmeasures

Geo

metricalreasoning

4.1 Ye

ar7

usecorre

ctlythe

vocabula

ry,notation

andlabe

lling

conventionsforlines,

anglesandshapes

identifyparalleland

perpend

icularlines;

knowthesumofangles

atapoin

t,onastraight

lineandinatriangle;

recognisevertically

opposite

angles

identifyalternate

anglesand

correspondingangles;

understandaproof

that:

tthea

nglesu

mofa

triangleis18

0and

ofaquadrila

teral

is360

tthee

xterior

angle

ofatriangle

is

equaltothe

sum

ofthetwoin

terior

oppositeangles



Year8

Year9

Year10

Year11

Extension

distinguishbetween

conventions,

definitionsandderived

properties

explainhowtofind,

calculateanduse:

tthes

umsofthe

interiorand

exteriorangles

ofquadrilaterals,

pentagonsand

hexagons

tthein

teriorand

exterioranglesof

regularpolygons

knowthedefinitionof

acircleandthenames

ofitsparts;explain

whyinscribedregular

polygonscanbe

constructedbyequal

divisionsofacircle

examineand

exam

ineandcreate

presentrigo

rousand

refinearguments

chainsofdeductive

sustainedarguments

insolutionsto

reasoninginsolutions

inthesolutionof

geometricalproblems,

tom

orecomplex

geometrical

problems;

distinguishingbetween

geometricalproblems

constructformal

practicaldemonstration

geometrical

proofs

andproof;produce

simpleproofs

usedynamicimagesto

examinethe

points

demonstrateinvariant

andlinesusedtocreate

relationshipsbetween

exam

ineandcreate

standardconstructions

radii,chordsand

proo

fsofthecircle

andusethe

conditions

tangentsincircles;

theo

rems;usecircle

ofcongruen

ceto

developarguments

theo

remstosolve

presentaproofthatthe

toexplainandjustify

prob

lems

standardconstructions

simplecircleproperties

areexact

andtheorems


28/42

30

Geo

metricalreasoning(continued)

4.1

Year8

Year9

Year10

Year7

identifyanduseangle,

sideand

symmetry

propertiesoftriangles

andquadrilaterals;

exploregeometrical

problem

sinvolving

theseproperties,

explainin

greasoning

orally,us

ingstep

by-stepdeduction

supporte

dbydiagrams

solvegeometrical

problemsusingside

andangleprope

rties

ofequilateral,

isoscelesandright-

angledtriangles

and

specialquadrilaterals,

explainingreaso

ning

withdiagramsandtext;

classifyquadrilat

erals

bytheirgeometrical

properties

knowthatiftwo

2-D

shapesarecongruent,

correspondingsides

andanglesareequal

solveproblems

usingpropertiesof

angles,ofparalleland

intersectinglinesand

oftrianglesandother

polygons,justifying

inferencesand

explainingreasoning

withdiagramsandtext

understand

congruenceand

exploresimilarity

investigatePythagoras

theorem,u

singavariety

ofmedia

,throughits

historicandcultural

roots

,includingpicture

proofs

solvegeometrical

problemsusing

propertiesoflines,

angles,polygons

andcircles;justify

argumentsand

solutionsusing

deductivereasoning

drawinferencesabout

propertiesofsimilar

2-Dshapesanduse

proportionalreasoning

tosolvegeometrical

andtrigonometrical

problems

visualiseandmanipulate

dynamicimagesand

usescaledrawingto

investigateareasof

squaresonsidesof

right-angledandnon

right-angledtriangles,

relatingfindingsto

Pythagorastheorem;

usePythagorastheorem

tosolveproblemsin2-D

andsimple3-Dcases

Year11

form

aliseexisting

knowledgeoflines,

anglesandpolygonsby:

tu

singthe

congruence

conditions(SSS,SAS,

RHS,ASA)todeduce

familiarproperties

oftrianglesand

quadrilaterals,e.g.

anisoscelestriangle

hastwoequal

angles

te

xplaining

whystandard

constructionswork,

e.g.observingthat

linesjoiningpoints

wherecompassarcs

meetaresidesofa

rhombus

Extension

(seeobjective

above

forprogression)

engagewithand

expl

ainthestagesof

avarietyofproofsof

Pyth

agorastheorem;

usePythagoras

theo

remtosolvemore

com

plex3-Dproblems

presentand

justify

aformalpro

ofof

Pythagoras

theorem




29/42

31

Geo

metricalreasoning(continued)

4.1 Ye

ar7

Year8

Year9

Year10

Year11

use2-Drepresentations

visualise3-Dsha

pes

tovisualise3-Dshapes

fromtheirnets;u

se

anddeducesomeof

geometricalproperties

theirpro

perties

ofcuboidsandshapes

madefromcubo

ids;

usesimpleplans

and

elevations

visualiseanduse2-D

representationsof3-D

objects;analyse3-D

shapesthrough2-D

projections,including

plansandelevations

visualiseanddescribe

propertiesofpoints,

linesandplanesin3-D

space,includingcross

sectionscreatedby

slicinga3-Dshape

visualiseand

manipulateimages

toestablish

trigonometrical

relationshipsby:

tgene

rating

trianglesusinga

rotatingunitradius

(circle,centrethe

origin)

tident

ifyingthe

propertiesof

similartriangles

formedby

enlargementsof

thecircle

usetrigonometrical

relationshipstosolve

simpleproblemsin2-D,

includingbearings

derivetheformula

ab

sinCforthe

area

ofatriangle;

usetrigonometrical

relat

ionshipsto

solvemorecomplex

2-Dproblemsand

prob

lemsin3-D,such

asth

eanglebetweena

lineandaplane

Extension

draw,sketch

and

comparethe

graphs

oftrigonometrical

functionsan

d

transformationsof

thesegraphs;provethe

sineandcos

inerules

andusethemtosolve

2-Dand3-D

problems

inarangeofcontexts




30/42

32

Tran

sformationsandcoordinates

4.2

Year8

Year9

Year10

Year11

Year7

understa

ndandusethe

languageandnotation

associate

dwith

reflections,translations

androtations

recogniseandvisualise

thesymm

etriesofa2-D

shape

transform

2-Dshapes

by:

trefle

ctingingiven

mirrorlines

trotat

ingabouta

give

npoint

ttrans

lating

explorethese

transform

ationsand

symmetriesusingICT

identifyallthe

symmetriesof2-

D

shapes

transform2-Dsh

apes

byrotation,refle

ction

andtranslation,on

paperandusing

ICT

tryoutmathema

tical

representationsof

simplecombinationsof

thesetransforma

tions

identifyreflection

symmetryin3-D

shapes

recognisethat

translations,rotations

andreflections

preservelengthand

angle,andmapobjects

ontocongruent

images

exploreandcompare

mathematical

representationsof

combinationsof

translations,rotations

andreflectionsof2-D

shapes,onpaperand

usingICT

deviseinstructionsfor

acomputertogenerate

andtransformshapes

usepreciselanguage

andnotation

todescribeand

generalisethe

resultsofcombining

transformationsof2-D

shapesonpaperand

usingICT,including:

trotat

ionsabout

anypoint

trefle

ctionsinany

line

ttrans

lationsusing

vectornotation

tatran

sformation

anditsinverse

generateandanalyse

patterns,e.g.Islamic

designs

expl

ainand

dem

onstrate

grap

hicallythe

effectsofcombining

translations,using

vectornotation,

inclu

ding:

tth

erulefor

additionofvectors

ts

calar

multiplicationofa

vector(repeated

addition)

Extension

explainand

demonstrate

graphicallyt

he

effectsofco

mbining

translations,using

vectornotat

ion,

including:

tthed

ifferenceof

twovec

tors

tther

esu

ltantof

twovec

tors

tthec

om

mutative

andassociative

propert

iesof

vectora

ddition

solvesimple

geometrical

problems

in2-Dusing

vectors




31/42

33

Tran

sformationsandcoordinates(continued)

4.2

Year8

Year9

Year10

Year11

Extension



Year7

useconv

entions

andnota

tionfor

2-Dcoordinatesin

allfourq

uadrants;

findcoordinatesof

pointsdetermined

bygeom

etrical

informat

ion

understandand

usethe

languageandno

tation

associatedwith

enlargement;en

large

2-Dshapes,givena

centreofenlarge

ment

andapositiveinteger

scalefactor;explore

enlargementusingICT

makescaledraw

ings

findthemidpoin

tof

thelinesegmentAB,

giventhecoordinates

ofpointsAandB

enlarge2-Dshapes,

givenacentreof

enlargementanda

positiveintegerscale

factor,onpaperand

usingICT;identify

thescalefactorof

anenlargement

astheratioofthe

lengthsofanytwo

correspondingline

segments;recognise

thatenlargements

preserveanglebutnot

length,andunderstand

theimplicationsof

enlargementfor

perimeter

enlarge2-Dshapes

usingpositive,

fractionalandnegative

scalefactors,on

paperandusing

ICT;usereciprocals

asamultiplicative

inverseinthecontext

ofenlargement;

recognisethesimilarity

ofresultingshapes

andexplaintheeffect

ofenlargementon

perimeter

useandinterpretmaps

andscaledrawings

inthecontextof

mathematicsandother

subjects

usethecoordinate

gridtosolveproblems

involvingtranslations,

rotations,reflections

andenlargements

applytheproperties

ofsimilartrianglesand

Pythagorastheorem

tosolvingproblems

presentedona2-D

coordinategrid;

usea3-Dcoordinate

gridtorepresent

simpleshapes

enlarge3-Dshapes;

iden

tifyandexplainthe

effectsofenlargement

ona

reasandvolumes

ofsimilarshapes

and

solids;relatethis

understandingto

prac

ticalcontexts,e.g.

inbiology


32/42

34

Con

structionandloci

4.3 Ye

ar7

Year8

Year9

Year10

Year11

Extension



usearulerand

protractorto:

tmeas

ureanddraw

linestothenearest

millimetreand

angles,including

refle

xangles,to

thenearestdegree

tcons

tructa

trian

gle,given

two

sidesandthe

inclu

dedangle

(SAS

)ortwoangles

and

theincluded

side

(ASA)

usestraightedgeand

compassestoconstruct:

tthem

idpoin

tand

perpendicular

bisectorofa

line

segment

ttheb

isector

of

anangle

tthep

erpend

icular

fromapointto

aline

tthep

erpend

icular

fromapointon

aline

tatria

ngle,given

threesides(SSS)

usestraightedgeand

compassestoconstruct

triangles,givenright

angle,hypotenuseand

side(RHS)

useICTtoexplore

constructionsof

trianglesandother

2-Dshapes

findthelocusofapoint

thatmovesaccording

toasimplerule,both

byreasoningandby

usingICT

usepropertiesof

2-Dand3-Dshapes

tomakeaccurate

constructionsonpaper

andusingICT;including

constructingtriangles

fromcombinations

ofsideandangle

facts,reviewingand

generalisingfindings

toidentifywhichof

theseconditionsdefine

uniqueconstructions

useICTtoexplore

useICTtoexplore

construc

tions

theseconstructions

useruler

andprotractor

toconstr

uctsimple

netsof3

-Dshapes,

cubo

id,regular

e.g.

tetrahed

ron,square-

basedpy

ramid,

triangula

rprism

findsimpleloci,both

byreasoningandby

usingICT,toproduce

shapesandpath

s,e.g.

anequilateraltriangle

visualiseanddescribe

thelocusofapoint

thatmovesaccording

toamorecomplex

rule;explainthe

pathusingaccurate

geometricalvocabulary

andnotationanduse

avarietyofmedia,

includingdynamic

geometrysoftware,

sketchesandgraphs

createachainof

reasoningtodeduce

theequationofa

circlebyapplying

Pyth

agorastheoremto

thelocusofapoint


33/42

35

Mea

suresandmensuration

4.4 Ye

ar7

Year8

Year9

Year10

Year11

Exte

nsion

choosea

nduseunits

ofmeasu

rementto

measure,estimate,

calculate

andsolve

problem

sineveryday

contexts

;convertone

metricunittoanother,

gram

stokilograms;

e.g.

readand

interpretscales

onarangeofmeasuring

instrume

nts

distinguishbetweenand

estimate

thesizeofacute,

obtuseandreflexangles

chooseanduse

units

ofmeasuremen

tto

measure,estim

ate,

calculateandsolve

problemsinar

angeof

contexts;know

rough

metricequivale

ntsof

imperialmeasu

resin

commonuse,s

uch

asmiles,pound

s(lb)

andpints

usebearingsto

specify

direction

solveproblemsinvolv

ing

measurementsina

varietyofcontexts;

convertbetweenarea

measures(e.g.mm2 to

cm2,cm2tom2,andvice

versa)andbetween

volumemeasures(e.g

.

mm3tocm3,cm3tom

3,

andviceversa)

Interpretandexplore

combiningmeasures

intoratesofchangein

everydaycontexts(e

.g.k

m

perhour,penceperme

tre);

usecompoundmeasures

tocompareinreal-life

contexts(e

.g.t

ravelgra

phs

andvalueformoney),

usingICTasappropriate

.

interpretanduse

compoundmeasures,

includingfromother

subjectsandreallife;

solveproblemsinvolving

rates;convertbetween

compoundmeasures,

choosingunitsmost

suitedtothesolution

makeconnections

betweenthecontinuity

ofthenumberlineand

continuousmeasures;

criticallyexaminethe

measurementsusedina

problemandtheireffect

ontheaccuracyofthe

solution,e.g.understand

howerrorscanbe

compounded

com

municate

thesolution

toa

problem

invo

lving

mea

surement,

explainingthe

limitationsof

accu

racyusing

upperand

lowe

rbounds




34/42

36

Mea

suresandmensuration(continued)

4.4 Ye

ar7

Year8

Year9

Year10

Year11

Extension

knowandusethe

deriveandusefo

rmulae

formulafortheareaof

fortheareaofa

arectangle;calculate

triangle,parallelo

gram

theperim

eterandarea

andtrapezium;

ofshapesmadefrom

calculateareasof

rectangles

compoundshapes

solveproble

ms

involvingmore

complexsha

pes

andsolids,in

cluding

segmentsof

circlesand

frustumsofcones

calculate

thesurface

areaofcubesand

cuboids

knowanduseth

e

formulaforthev

olume

ofacuboid;calculate

volumesandsur

face

areasofcuboids

and

shapesmadefro

m

cuboids

knowandusethe

formulaeforthe

circumferenceandarea

ofacircle

calculatethesurface

areaandvolumeof

rightprisms

presentaconcise

reasonedargumentto

deriveformulaefor:

tlengt

hsofcircular

arcs

tareas

ofsectorsof

acircle

tsurfa

ceareaof

acylinder

tvolum

eofa

cylinder

solveproblems

involvingtheuseof

theseformulae

pres

entaconcise

reasonedargument

whe

nderiving

form

ulaeforthe

surfaceareasof

pyra

midsandcones;

expl

oreconnections

betw

een:

tfo

rmulaefor

thevolumeofa

pyramidandthe

relatedcuboid

tfo

rmulaefor

thesurfacearea

andvolumeofa

sphereandthe

circumscribedand

inscribedcubes

solveproblems

invo

lvingtheuseof

thes

eformulae




35/42

37

5Statistics

5.1

Spec

ifyingaproblem,planningandcollecting



Year7

Year8

Year9

Year10

Year11

Extension

suggestpossible

answers,givena

question

thatcanbe

addresse

dbystatistical

methods

discussaproblemthat

canbeaddressedby

statisticalmetho

ds

andidentifyrelated

questionstoexp

lore

suggestaproblemto

exploreusingstatistical

methods,frame

questionsandraise

conjectures

independentlydevisea

suitableplanforamore

complexstatistical

project,selecting

suitablehypothesesto

addresstheproblem

evaluatepossible

difficultieswith

plan

nedapproaches;

adju

sttheproject

plan

accordingly,

inclu

dingreconsidering

hypotheses

decidew

hichdata

wouldberelevanttoan

enquiryandpossible

sources

decidewhichdata

tocollecttoansw

er

aquestionandthe

degreeofaccura

cy

needed;identify

possiblesources;

considerappropriate

samplesize

discusshowdifferent

setsofdatarelateto

theproblem;identify

possibleprimaryor

secondarysources;

determinethe

samplesizeandmost

appropriatedegreeof

accuracy

justifythesampling

methodselected,

identifypossible

sourcesofbiasand

planhowtominimiseit

iden

tifypractical

prob

lemssuchas

non-responseor

miss

ingdataandrefine

approachestominimise

theirimpactonthe

valid

ityoftheresults


36/42

38

5.1

Spec

ifyingaproblem,planningandcollecting(continued)

Year7

Year8

Year9

Year10

Year11

Extension

planhow

tocollectand

organise

smallsetsof

datafrom

surveysand

experiments:

tdesig

ndata

colle

ctionsheets

orquestionnaires

touseinasimple

surv

ey

tcons

truct

frequencytables

forg

athering

discretedata,

grou

pedwhere

appropriate

ineq

ualclass

intervals

planhowtocollect

thedata;constru

ct

frequencytables

with

equalclassintervalsfor

gatheringcontin

uous

dataandtwo-wa

y

tablesforrecording

discretedata

designasurveyor

experimenttocapture

thenecessarydata

fromoneormore

sources;design,trial

andifnecessaryrefine

datacollectionsheets;

constructtablesfor

gatheringlargediscrete

andcontinuoussets

ofrawdata,choosing

suitableclassintervals;

designandusetwo-

waytables

gatherdatafrom

specifiedsecondary

sources,including

printedtablesandlists,

andICT-basedsources,

includingtheinternet



decideonthebest

methodsfortesting

thehypotheses;

select,justifyanduse

thedata-gathering

techniquemost

appropriatetothe

context,deciding

betweenarange

ofsources:primary

(observation,controlled

experiment,data

logging)andsecondary

(spreadsheetdata,

printedtables,lists)

select,justifyanduse

thedata-gathering

tech

niqueappropriate

toco

mplexand

unfa

miliarproblems,

iden

tifyingpotential

barriersandlimitations;

iden

tifywhatextra

informationmaybe

requ

iredtopursuea

furth

erlineofenquiry

selectandcritically

evaluateasampling

schemeand

amethod

toinvestigatea

population,

including

randomand

stratified

sampling;ex

plainthe

effectonreliability

andvalidity

collectsmallsetsof

collectdatausinga

datafrom

surveys

suitablemethod

(e.g.

andexperiments,as

observation,con

trolled

planned

experiment,data

loggingusingICT)


37/42

39The National Strategies | SecondarySecondary mathematics subject leader development materials: Spring 2010

Processingandrepresenting

data

5.2 Y

ear7

Year8

Year9

Year10

Year11

Extension

calculate

statisticsfor

smallset

sofdiscrete

data:

tfindt

hemode,

med

ianandrange,

and

themodal

classforgrouped

data

tcalcu

latethe

mea

n,including

from

asimple

frequencytable,

usingacalculator

fora

largernumber

ofitems

calculatestatistics

forsetsofdiscrete

andcontinuous

data,includingw

ith

acalculatorand

spreadsheet;recognise

whenitisappropriate

tousetherange,mean,

medianandmod

eand,

forgroupeddata

,the

modalclass

calculatestatistics

andselectthosemost

appropriatetothe

problemorwhich

addressthequestions

posed

useanappropriate

rangeofstatistical

methodstoexplore

andsummariselarge

datasets,justifying

thechoicesmade;

includegrouping

data,estimatingand

findingthemean,

median,quartilesand

interquartilerange

proc

essdatadrawn

from

problems

invo

lvingseasonality

and

trendsinatime

serie

s;chooseand

com

binestatistical

methodstoanalyse

theproblem,including

mov

ingaverages



38/42

Processingandrepresenting

data(continued)

5.2 Y

ear7

Year8

Year9

Year10

construc

t,onpaper

andusingICT,graphs

anddiag

ramsto

represen

tdata,

including:

tbar-l

inegraphs

tfrequ

ency

diag

ramsfor

grou

peddiscrete

data

tsimp

lepiecharts

constructgraphical

representations,

on

paperandusing

ICT,

andidentifywhich

aremostusefulinthe

contextoftheproblem,

including:

tpiec

hartsfo

r

categoricaldata

tbarc

hartsand

frequency

diagramsfor

discreteand

continuousdata

tsimp

lelineg

raphs

fortimeseries

tsimp

lescatter

graphs

tstem-and-le

af

diagrams

select,construct

andmodify,on

paperandusingICT,

suitablegraphical

representationsto

progressanenquiry

andidentifykey

featurespresentinthe

data.Include:

tlineg

raphsfor

timeseries

tscatt

ergraphsto

developfurther

understandingof

correlation

constructonpaper

andusingICT

suitablegraphical

representations,

including:

thisto

grams

forgrouped

continuousdata

withequalclass

intervals

tcumu

lative

frequencytables

anddiagrams

tboxp

lots

tscatt

ergraphsand

linesofbestfit

(byeye)

justifytheirsuitability

withreferencetothe

contextoftheproblem

andtheaudience

workthroughtheentire

handlingdatacycleto

explorerelationships

withinbi-variatedata

,

includingapplications

toglobalcitizenship

,e.g.

howfairisoursociety?

Year11

Extension

chooseandcombine

suita

blegraphical

representationsto

prog

ressanunfamiliar

ornon-routineenquiry,

inclu

dinghistograms

with

equalorunequal

class

intervals

usepreciseand

consistentgraphical

representati

onto

progressan

unfamiliar

andnon-rou

tine

enquiry

40 The National Strategies | SecondarySecondary mathematics subject leader development materials: Spring 2010



39/42

41

Inte

rpretinganddiscussing

results

5.3

Year8

Year9

Year10

Year11

Extension



Year7

interpret

diagramsand

graphs(includingpie

charts)anddrawsimple

conclusionsbasedon

theshap

eofgraphs

andsimp

lestatisticsfor

asingledistribution

compare

twosimple

distributionsusingthe

rangean

doneofthe

mode,m

edianormean

writeashortreportof

astatisticalenquiry,

includingappropriate

diagrams,graphsand

charts,usingICTas

appropriate;justifythe

choiceofpresentation

interprettables,

graphsanddiagrams

fordiscreteand

continuousdata,

relatingsummar

y

statisticsandfindings

tothequestions

being

explored

comparetwo

distributionsusingthe

rangeandoneormore

ofthemode,me

dian

andmean

writeaboutand

discusstheresultsofa

statisticalenquir

yusing

ICTasappropriate;

justifythemetho

ds

used

interpretgraphsand

diagramsandmake

inferencestosupport

orcastdoubtoninitial

conjectures;havea

basicunderstandingof

correlation

comparetwoor

moredistributions

andmakeinferences,

usingtheshapeof

thedistributionsand

appropriatestatistics

reviewinterpretations

andresultsofa

statisticalenquiryon

thebasisofdiscussions;

communicatethese

interpretationsand

resultsusingselected

tables,graphsand

diagrams

findpatternsand

interpretandcompare

explainand

justify

exceptionsandexplain

distr

ibutions,including

assumptionsand

anomalies;including

cum

ulativefrequency

constraints;include

interpretationof

diag

rams;makeand

interpretatio

nand

socialstatisticsand

discussinferences,

comparison

of

evaluationofthe

usingtheshapeof

histogramswith

strengthofassociation

thedistributionsand

unequalclas

sintervals

withinbi-variatedata

mea

suresofaverage

(correlation,linesof

and

spread,including

bestfit)

med

ianandquartiles

evaluatetheresults

criticallyexamine

usestatistical

ofastatistical

strat

egiesadopted

analysiseffe

ctivelyin

enquiry;reviewand

and

arguments

presentingc

onvincing

justifyorrefinethe

pres

ented,relating

conclusions;critically

choiceofstatistical

them

totheoriginal

reflectonow

nlines

representationsand

hypotheses;recognise

ofenquiry;s

earch

relatesummariseddata

thelimitationsofany

forandappr

eciate

tothequestionsbeing

assu

mptionsandthe

moreelegan

tforms

explored

effectsthatvarying

ofcommunicating

assu

mptionscould

conclusions

have

onconclusions

draw

nfromdata

analysis


40/42

42 The National Strategies | SecondarySecondary mathematics subject leader development materials: Spring 2010

Probability

5.4 Y

ear7

Year8

Year9

Year10

Year11

Extension

usevoca

bularyand

ideasofprobability,

drawing

onexperience

interprettheresults

ofanexperiment

usingthelangua

geof

probability;appreciate

thatrandompro

cesses

areunpredictable

interpretresults

involvinguncertainty

andprediction

identifywhenthe

eventsinaproblem

aremutuallyexclusive

orindependent;use

andinterprettree

diagramstorepresent

outcomesofcombined

eventsandtoinform

thecalculationof

theirprobabilities;

decidewhentoadd

andwhentomultiply

probabilities

interprettheeffect

onp

robabilityof

cont

extsinvolving

selectionwithand

with

outreplacement;

chooseandcombine

representations

toco

mmunicate

prob

abilitiesaspartof

asolutiontoaproblem

recognisewhenand

howtowork

with

probabilitiesassociated

withindependent

andmutually

exclusiveeventswhen

interpreting

data

understa

ndanduse

theprob

abilityscale

from0to

1;findand

justifypr

obabilities

basedon

equallylikely

outcome

sinsimple

contexts

;identifyall

thepossiblemutually

exclusive

outcomesofa

singleev

ent

knowthatifthe

probabilityofan

event

occurringispthenthe

probabilityofitnot

occurringis1p

;use

diagramsandtablesto

recordinasystematic

wayallpossible

mutuallyexclusive

outcomesforsin

gle

eventsandfortw

o

successiveevent

s

identifyallthemutually

exclusiveoutcomes

ofanexperiment;

knowthatthesum

ofprobabilitiesofall

mutuallyexclusive

outcomesis1and

usethiswhensolving

problems



41/42

43

Probability(continued)

5.4 Y

ear7

Year8

Year9

Year10

Year11

Extension

estimate

probabilities

bycollec

tingdatafrom

asimple

experiment

andreco

rdingitin

afrequencytable;

compare

experimental

andtheo

retical

probabilitiesinsimple

contexts

compareestimated

experimental

probabilities

withtheoretical

probabilities,

recognisingthat:

tifan

experim

ent

isrepeatedthe

outcomemay,and

usuallywill,be

different

tincre

asingthe

numberoftimes

anexperime

ntis

repeatedge

nerally

leadstobetter

estimatesof

probability

compareexperimental

andtheoretical

probabilitiesinarange

ofcontexts;appreciate

thedifference

betweenmathematical

explanationand

experimentalevidence

userelativefrequency

asanestimateof

probability,including

simulationusingICT

togeneratelarger

samples;discuss

itsreliabilitybased

onsamplesizeand

usetointerpretand

compareoutcomesof

experiments

expl

orearelevantand

purp

osefulproblem

invo

lvinguncertainty;

estim

ateriskby

mod

ellingrealevents

throughsimulation;

justifydecisionsbased

one

xperimental

prob

abilityand

com

mentonthe

effectofassumptions

and

samplesize

onthereliabilityof

conc

lusions




42/42

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