Unit 2: Axioms and arrays - Oulder Hill

Unit2:AxiomsandarraysWeek1Lesson1:Whatismultiplication?

1. Usethebarmodeltocopyandcompletethecalculationsinthefollowingfactfamily:

2. Copyandcompletethebarmodelandfactfamilybelow:

3. Drawthediagrambelowandcompletethefactfamilythatitrepresents.

4. a)Writethefourdifferentcalculationsinthefactfamilyof5 × 6b)Drawthreedifferentdiagramstorepresent5 × 6

ConceptCorner

Copyandcomplete:

Divisionisthe________operationofmultiplication.

Inamultiplication/divisionfactfamily,thereare_______relatedcalculations.

Thediagrambelowshowsthat:

4 × 3 = 12 12 ÷____ = 3

3 ×_____ = 12 12 ÷_____ = 4

3

15 5 × 3 = _____

15 ÷_____ = 3

3 ×_____ = _____

15 ÷_____ = 5

______ × 4 = ______

80 ÷______ = ______

______ × ______ = ______

______ ÷ ______ = ______

______ × 3 = ______

______ ÷ 7 = ______

______ × ______ = ______

______ ÷ ______ = ______

four3 4 4

inverse

Unit2:Axiomsandarrays

5. Foreachproblem,drawamodeltorepresentandcalculatethesolution.

6. CopyandcompleteaFrayermodeltodescribemultiplication

Examplesandnon-examples

Modelsandimages

Factsandcharacteristics

Definition

Questionsfordepth

1. Giventhat2.5 × 6.4 = 16.Findthethreeremainingcalculationsinthefactfamily.

2. Writeatleastfourdifferentwordproblemsinvolvingmultiplicationordivisionthat

thismodelcouldrepresent.

Multiplication

2.5

15

Abagcontains7sweets.Thereare9bags. Howmanysweetsarethereintotal?

Abagcontains9sweets.Thereare7bags. Howmanysweetsarethereintotal?

Abagcontains9sweets.Thereare63sweetsintotal.Howmanybagsofsweetsarethere?

Abagcontains7sweets.Thereare63sweetsintotal.Howmanybagsofsweetsarethere?

Thereare63sweetsintotal.Theyareputinto9bags. Howmanysweetsineachbag?

Thereare63sweetsintotal.Theyareputinto7bags. Howmanysweetsineachbag?


Week1Lesson2:Whatiscommutativity?

1. Jenniehasdrawnanarray,shehasgroupedthedots

toshow3 × 6 = 18

a) Drawthearray,groupthedotsdifferentlytoshow6 × 3 = 18.

b) Bygroupinganarrayintwodifferentways,showthat4 × 3 = 3 × 4.

c) Whatothercalculationscanyoushowusinga4 × 3array? 2. a)Findtwodifferentwaystocompletethiscalculation.

32 =×

b) Howmanyotherwayscanyoufillintheblanksinthecalculationusingpositiveintegers?

3. Zarabuys20redapplesfor23peach.Charlottebuys23greenapplesfor20peach.

a) Howmuchdideachpersonspend?

b) IfZarabought40redappleshowmanygreenappleswouldCharlottehaveto

buytospendthesame?

ConceptCorner

Copyandcomplete:

Ifanoperationiscommutativethenwecanapplytheoperationtotwonumbersin

any____________.

Forexample,additionandmultiplicationare_____________________:

e.g. 𝟒 +____ = 𝟕 + 𝟒 𝟑 × 𝟓 = 𝟓 ×____

7 4

7 4 5

3

3

5

order 7 3commutative


4. Completethecalculationsbelowtocreatethreedifferentfactfamilies.Drawadiagramtorepresenteachone

5. Copyandcompletethefollowingmultiplicationgrids.

a) b)

Questionsfordepth:

1. Akiranoticesaninterestingpattern:

a) Whatisthenextequationinthelist?

b) Writeasentencetodescribethepattern.

c) Explainwhythispatternwillalwayswork. 2. Tomwantstobuysinglescoopoficecreamwithonetopping.

a) DoesTomhavemoreoptionsinAlice’sorBill’s?Justifyyouranswer.b) IfinsteadTomwantedtwoscoopsandonetopping,whichwouldgivehimmore

choice?Justifyyouranswer.

× 11

3 27

25 65

66

40

× 4 9

2 10

12 21

11

60

____ × ____ = 100

100 ÷ ____ = ____

____ × ____ = ____

____ ÷ ____ = ____

Alice’s Ice Cream Parlour

Flavours: Chocolate or Vanilla

Toppings: Nuts, Chocolate Sauce, Sprinkles or a Flake

Bill’s Ice Cream Shop

Flavours: Chocolate, Vanilla, Strawberry or Coffee

Toppings: Nuts or a Flake.

1 + 1 + 1 = 3 2 + 2 + 2 = 3 + 3

3 + 3 + 3 = 3 + 3 + 3

4 + 4 + 4 = 3 + 3 + 3 + 3

5 + 5 + 5 = 3 + 3 + 3 + 3 + 3


Week1Lesson3:MultiplicationandDivision

1. a) Explainhowthefollowingmodelsshowthat6 × 8 = 8 × 6

b) Drawtwosimilarmodelstoshowthefollowing:i. 2 × 6 = 6 × 2ii. 5 × 7 = 7 × 5

2. Explainhowthetwodivisionproblemslinktothemultiplicationfact6 × 4 = 24

i. 24biscuitsaresharedamongst4people.Howmanybiscuitsdoeseachpersonreceive?

ii. 24biscuitsareshared,eachpersonreceives4biscuits.Howmanypeoplearethere?

Writesimilarpairsofdivisionproblemsforthefollowingmultiplicationfacts:iii. 12 × 6 = 72iv. 150 × 6 = 900v. D

E× 8 = 4

ConceptCorner

Copyandcomplete:

Divisioncanbeinterpretedindifferentways.

Forexample,24 ÷ 6 = 4canbeunderstoodas24dividedinto6________groupsoras24dividedinto_________of6.

Understandingthatmultiplicationiscommutativeallowsustobeflexiblewiththewaywethinkaboutdivision.Knowingthat:

21isequalto___groupsof3

canbeusedtoworkout:

___dividedinto3equalgroups

equalgroups7 21

6 6 6 6 6 6 6 6

8 8 8 8 8 8


3. Lookatthefollowingtableandwriteastatementforeachemptycell.

4. Writeatleastfourdifferentwordproblemsinvolvingmultiplicationordivisionthat

thismodelcouldrepresent.

Questionsfordepth

1. Explainhowtouse:a. 400 ÷ 25 = 16toworkout384 ÷ 16b. 221 ÷ 17 = 13toworkout234 ÷ 13c. 322 ÷ 14 = 23toworkout299 ÷ 23d. Createsomeofyourownexamplesofthisstyleofquestion

2. CopyandcompleteaFrayermodeltodescribedivision

Examplesandnon-examples

Modelsandimages

Factsandcharacteristics

Definition

3 × 4 Threepacksoffourpensis12pensintotal

Therearethreepensperpack.Therearefourpacksso12pensintotal

4 × 3 Therearefourpensperpack.Therearethreepacksso12pensintotal

a)

12 ÷ 3 Thereare12pensinthreepackets.Eachpacketcontains4pens

b)

12 ÷ 4 c) d)

9

54

Division


Week1Lesson4:WhatisAssociativity?

1. Evaluatetheexpressionsineachpairtoshowthattheyareequal:

a) (3 × 8) × 5and3 × (8 × 5)

b) (18 × 2) × 5and18 × (2 × 5)

c) (2.5 × 4) × 2and2.5 × (4 × 2)

2. Sallyuses48cubestomakeacuboid.Shebreaksupthecuboidinfourdifferentways.Foreachimage,copyandcompletethecorrespondingcalculation.

a) b) c) d)

2 × (___ × ___) 4 × (___ × ___) ___ × (4 × ___) (___ × ___) × ___

ConceptCorner

Copyandcomplete:

Thesecalculationsanddiagramsshowhow____________hasbeenusedtoshowthat

5 × 12 = 15 × 4

12

5

𝟓 × 𝟏𝟐 = 𝟓 × (___ × 𝟒) = (𝟓 × ___) × 𝟒 = ___ × 𝟒

15

4

5 5 5

___ 4

4

___

4

associativity 31535 4


3. Completethecalculationstomatchthediagrams:

4. Copyandcompletethecalculations:

a) 16 × 5 = (8 × ___) × 5 = 8 ×(___ × 5) = 8 × ___ = ___

b) 16 × 35 = 16 × (5 × ___) = (16 × ___) × ___ = ___ × ___ = ___

c) 25 × 6 = 25 × (___ × 3) = (25 × ___) × 3 = ___ × 3 = ___

d) 25 × 12 = 25 × (___ × ___) = (25 × ___) × ___ = ___ × ___ = ___

e) 8 × 35 = 8 × (___ × ___) = (8 × ___) × ___ = ___ × ___ = ___

f) 1.6 × 35 = 1.6 × (___ × ___) = (1.6 × ___) × ___ = ___ × ___ = ___

5. Usethefactthat32 × 28 = 896toworkoutthefollowingcalculations:

a) 10 × 32 × 28 b) 32 × 280 c) 64 × 28

d) 16 × 56 e) 3.2 × 2800 f) 160 × 56

Questionsfordepth

1. Foreachdiagramusethegroupstowriteacalculation.Thefirsthasbeencompleted.

2. Createasimilardiagramtothequestionaboveforthe24dots.Finddifferentwaysof

writing24usingthedifferentgroupingsofdots.

e.g.

12 × 3 = 36

a)

b)

c) d) e)

36

5 15

12

5 5 5

___ 12

12

___

12

5 × 36 = 5 × (___ × 12) = (5 × ___) × 12 = ___ × 12


Week2Lesson1:Whatisthedistributiveproperty?

1. Foreachdiagramwritedownthecorrespondingequation.Thefirstonehasbeendoneforyou.a) b)

4 × (2 + 4) = 4 × 2 + 4 × 4

c) d)

e) f)

ConceptCorner

Belowisacalculationanddiagramtoshowthedistributivityofmultiplicationoveraddition.

Usetheworkedexampletocopyandcompletetheotherdiagramandcalculation.

e.g.

8 × 14 = 8 × (10 + 4) 9 × 27 = ___ × (___ + ___)

= 8 × 10 + 8 × 4 = …

= 80 + 32 =

= 112

9

10 5

10

10 2.5

2.5

2 6

10

5.5 4.5

20

9

7 10 4

8 80 32


2. Completethedifferentwaysofusingdistributivityforcalculating5 × 17:

a) 5 × 17 = 5 × (10 + ___) = 5 × 10 + 5 × ___ = 50 + ___ = ___

b) 5 × 17 = 5 × (8 + ___) = 5 × 8 + 5 ×___ = 40 + ___ = ___

c) 5 × 17 = 5 × (5 +___) = 5 × 5 + 5 ×___ = ___ + ___ = ___

d) 5 × 17 = 5 × (20 − ___) = 5 × 20 − 5 × ___ = ___ − ___ = ___

e) Whichwayisyourpreferredmethod?Why?

3. Usethedistributivepropertytocalculateeachoftheseintwodifferentways.Writeequationstoshoweachstepofyourcalculationstrategy.

a) 5 × 24 b)11 × 11 c)8 × 99 d)2.5 × 12

4. Ariannawantstocalculate5 × 128.Shedrawsthisdiagramtohelpher:

a) Writeacorrespondingcalculationrepresentedbythearrayandfindasolution.b) Useassociativitytohelpyoutocalculate5 × 128inadifferentway.c) Drawadiagramtoshowhowdistributivitycanbeusedtocalculate6 × 1328.

Questionsfordepth

1. Priyahasamethodformultiplyinganumberby999.

a) UsePriya’smethodtocalculate7 × 999

b) UsePriya’smethodtocalculate999 × 999

c) Drawadiagramtoshowwhythismethodwillalwayswork.

2. AdaptPriya’smethodtohelpyoutocalculate:

a) 9999 × 9999

b) 99999 × 99999

c) 999999 × 999999

5

100

20

8

Ifirstmultiplythenumberby1000,thentakeawaythe

originalnumber.


Week2Lesson2:MultiplicationTables

1. Copyandcompletethecalculationsbelow:

a) 2 × (5 × 4) = ____ b) 7 × 8 = 8 × ___ c) 9 × 4 = 9 × (2 × ___) d) 64 = 4 × ___ × 8 e) ____ = 4 × 12 + 8 × 12 f) 12 × 6 = 2 × (___ × 6) g) 6 × ___ + 5 × ___ = 7 × (6 + 5) h) 88 = 4 × (___ × ___)

2. Giveanexampletoshoweachofthefollowingstatementsaretrue.Anexamplehasbeendoneforyou:e.g.IfIknowmy10and2timestablesthenIcanfindmy8timestables...…forexample,tofind8 × 7Icansubtract2 × 7from10 × 7

a) IfIknowmy2timestableIcanworkoutmy4or8timestablesb) Icanfindmy9timestablefrommy10timestablec) Icanfindmy6timestableusingthe5timestabled) Icanworkoutmy7timestableifIknowmy5and2timestables

ConceptCorner

Forexample,associativitycanbeusedtoexplainwhyvaluesinthe4timestablescanbefoundby________________thevaluesinthe___timestables.

(3 × 2) × 2 = 3 × (2 × 2) = 3 × ___

Distributivityshowstherelationshipbetweenthe2,10and12timestables.

7 × 12 = 7 × (___ + 2) = 7 × 10 + ___ × 2

Copyandcomplete:

Understandingoftheaxiomscanbeusedtoexplainandunderstandrelationshipsbetween__________________facts.

2 timestables

4 doubling

10

7


3. Somenumbersappearinmanyofthe1-12timestables.Forexample,16appearsinthe1,2,4and8timestables.Lookatthenumbersbelow.Puttheminorderforhowmanyofthe1-12timestablestheyappearin(leasttomost).

243236

4. Giventhat18 × 13 = 234,findthesolutionstothefollowingcalculations:

5. Decideifthestatementsbelowaretrueorfalse,withexamplestoshowwhenfalse.

a) Everynumberinthe4timestableisalsointhe2timestable.

b) Everynumberinthe8timestableisalsointhe12timestable.

c) Everynumberinthe12timestableisalsointhe3timestable.

d) Everynumberinthe3timestableisalsointhe6timestable.

e) Everyevennumberinthe5timestableisalsointhe10timestable.

Questionsfordepth

1. Thestatementsbelowrefertohowmanytimesnumbersappearina1-12timestablesgrid.Explainwhytheyarealltrue:a) Primenumbersappearanevennumberoftimes.

b) Squarenumbersappearanoddnumberoftimes.

c) 24appearsmoretimesthananyothernumber.

2. Thestatementsbelowrefertotimestablesthatcontinuetoinfinity.Decideifthestatementsbelowaretrueorfalse.a) Onlyhalfofthenumbersinthe6timestableareinthe3timestable.

b) Onequarterofthenumbersinthe12timestableareinthe4timestable.

c) Halfoftheevennumbersinthe3timestableareinthe12timestable.

d) Oneseventhofthenumbersinthe7timestablearemultiplesof3.

3. Createyourownstatementsthatcouldbeincludedinquestions1&2.

a) 9 × 13 b) 36 × 13 c) 9 × 26 d) 19 × 13

e) 28 × 13 f) 18 × 6.5 g) 9 × 6.5 h) 13 × 18


Week2Lesson3:Numberpyramids

1. Thebricksinthemiddlerowofthepyramidsarecompletedbymultiplyingthenumbersinthetwobricksbeneaththem.Thetopbrickiscompletedbysummingthenumbersinthemiddlerow.Copyandcompletethepyramidsbelow.Thefirstonehasbeendoneforyou.a) b) c)

d) e) f)

2. Thepyramidsbelowhavethesamerulesasthepyramidsinquestion1.a)b)

Completethepyramidsinasmanydifferentwaysasyoucanusingpositiveintegers.Whatdoyounoticeaboutsumoftheouterbricksinthebottomrowforeachsolution?Howisthisconnectedwiththenumberinthetopbrick?

ConceptCorner

multiplication𝑎

Copyandcomplete:

Thedistributivepropertytellsusanimportantrelationshipthatinvolves______________,additionandsubtraction.

Wecanrepresentthedistributivepropertyusingdiagramsandcalculations.Wedon’tneedtoknowthevaluesinthecalculations,andcanshowtherelationshipusing_____________:

𝑎 × (𝑏 + 𝑐) = ___ × 𝑏 + 𝑎 × 𝑐

___

𝑎

𝑐

𝑏

2 3 4

6 12

18

7 5 6

2 9

12

12 2

36

3

30 50

8

24

72

7

49 6

36


3. Thediagramsbelowcanbeusedtoshowthedistributiveproperty.Foreachdiagramwriteouttwoequivalentexpressions.Thefirsthasbeendoneforyou.

a) 2 × (6 + 3) = 2 × 6 + 2 × 3

b) c)

d) e) f)

4. Matchthecalculationsbelowintopairs.

Questionsfordepth

1. Labelthediagramsandwriteexpressionsthateachcouldrepresenta)

b)

c) d)

6

2

3 7

4

2.5 3

1.2

4

𝑎

𝑐

𝑏 𝑎

𝑏

𝑐 𝑎

4

𝑏

𝑐 × 𝑎 𝑎 × 𝑏

6 × (7 + 8) 7 × 8 + 6 × 8

(6 + 8) × 7

6 × 7 + 8 × 6

8 × (7 + 6)

7 × 14

𝑏 × 𝑐 𝑎 × 𝑏

6 × 𝑏 6 × 𝑎 4 × 𝑏 6 × 𝑎


Week2Lesson4:Numbertalks

1. 𝟐𝟒 × 𝟗hasbeencalculatedinthreedifferentways.Copyandcompletethecalculations.Foreach,drawadiagramtoshowthecalculationandstatewhichaxiomhasbeenused.

a) 24 × 9 = 9 × (20 + ___) = ___ × 20 + 9 × 4 = ___ + 36 = 216

b) 24 × 9 = (2 × ___) × 9 = ___ × (12 × 9) = 2 × ___ = 216

c) 24 × 9 = 24 × (___ − 1) = 24 × 10 − 24 × ___ = 240 − ___ = 216

d) 24 × 9 = (4 × 6) × 9 = ___ × (6 × 9) = 4 × ___ = 216

e) 24 × 9 = 9 × (10 + 10 + ___) = ___ × 10 + ___ × 10 + 9 × ___ = 90 + ___ + ___ = 216

2. Withoutcalculatingtheanswer,writeoutthecalculationwitheither<,>or=betweenthemtomakestatementscorrect:

14 × 5 14 × 6

24 × 9 23 × 8

18 × 5 9 × 10

17 × 7 16 × 8

ConceptCorner

Copyandcomplete:

Tocalculate18 × 5,Icoulduse____________torewritethecalculationas:

(9 × ___) × 5 = 9 × (2 × 5) = 9 × ___ = 90

Icouldalsousethedistributiveandcommutative__________torewritethecalculationas:

5 × (10 + ___) = 5 × 10 + 5 × 8 = 50 + ___ = 90

Oras:

5 × (___ − 2) = 5 × 20 − 5 × 2 = ___ + 10 = 90

a) 4, 9, 180 b) 12, 2, 108

associativity

2

20 10 40

8 property 100


3. Evaluateeachofthefollowingbyfirstapplyingeitherthedistributiveorassociativeaxioms.Thefirstexamplehasbeendoneforyou:

a) 12 × 25

b) 9 × 25

c) 37 × 11

d) 99 × 37

e) 128 × 13

f) 25 × 16 × 8

4. Decideonanefficientcalculationmethodandcomplete,showingthestepsbywritingcalculationsand/ordrawingamodel.

a) 13 × 25

b) 25 × 19

c) 24 × 11

d) 6 × 24

e) 144 × 15

f) 20 × 12 × 15

Questionsfordepth

1. Rewriteeachcalculationbelowbyapplyingthedistributiveaxiom.Usethistoevaluateeachone.a) 8 × 2.5 + 2 × 2.5 b)93 × 6.5 + 7 × 6.5

c) 21 × 0.3 + 79 × 0.3 d) 113 × 1.87 − 1.87 × 13

2. Foreachofthefollowingstatements,decideiftheyaretrueorfalse.Justifyyouranswerandgiveanexample.

a) Twonumberscanbemultipliedinanyorder

b) Alloperationsarecommutative

c) ∎× (⊿ +∗) = (⊿ +∗) × ∎

d) ∎− (⊿ −∗) = (∎ − ⊿) −∗

e) ∎÷ (⊿ +∗) = ∎ ÷ ⊿ + ∎÷∗

f) Multiplicationisdistributiveoveradditionandsubtraction

“Icanusetheassociativeaxiomtorewrite12 × 25as:

(3 × 4) × 25 = 3 × (4 × 25) = 3 × 100 = 300”

Unit 2: Axioms and arrays - Oulder Hill

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Transcript of Unit 2: Axioms and arrays - Oulder Hill