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Breuke - Vermenigvuldiging van breuke

Module by: Siyavula Uploaders. E-mail the author

WISKUNDE

Gewone Breuke

OPVOEDERS AFDELING

Memorandum

18.1

OPTELLING

1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} + 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} + 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {}

3737 size 12{ { { size 8{3} } over { size 8{7} } } } {} + 3737 size 12{ { { size 8{3} } over { size 8{7} } } } {}

2323 size 12{ { { size 8{2} } over { size 8{3} } } } {} + 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {} + 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {}

2525 size 12{ { { size 8{2} } over { size 8{5} } } } {} + 2525 size 12{ { { size 8{2} } over { size 8{5} } } } {} + 2525 size 12{ { { size 8{2} } over { size 8{5} } } } {} + 2525 size 12{ { { size 8{2} } over { size 8{5} } } } {}

PRODUK

2 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

1 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

6 7 6 7 size 12{ { { size 8{6} } over { size 8{7} } } } {}
(1)

2

1 3535 size 12{ { { size 8{3} } over { size 8{5} } } } {}

b) Getallelyn / Teller x Teller

Noemer x Noemer

d)

(i) 21102110 size 12{ { { size 8{"21"} } over { size 8{"10"} } } } {}

= 2 110110 size 12{ { { size 8{1} } over { size 8{"10"} } } } {}

(ii) 123123 size 12{ { { size 8{"12"} } over { size 8{3} } } } {}

= 4

(iii) 849849 size 12{ { { size 8{"84"} } over { size 8{9} } } } {}

= 9 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {}

Figure 1
Figure 1 (graphics1.png)

19.1

a) 1

b) 1

c) 1

d) 1

19.2 Produk is elke keer 1

19.4 a) 20172017 size 12{ { { size 8{"20"} } over { size 8{"17"} } } } {}

b) 140140 size 12{ { { size 8{1} } over { size 8{"40"} } } } {}

c) 531531 size 12{ { { size 8{5} } over { size 8{"31"} } } } {}

d) 873873 size 12{ { { size 8{8} } over { size 8{"73"} } } } {}

19.5 c) 531531 size 12{ { { size 8{5} } over { size 8{"31"} } } } {} : Maak eers onegte breuk ( 315315 size 12{ { { size 8{"31"} } over { size 8{5} } } } {} )

d) 873873 size 12{ { { size 8{8} } over { size 8{"73"} } } } {} : Maak eers onegte breuk ( 738738 size 12{ { { size 8{"73"} } over { size 8{8} } } } {})

20. a) 1 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {} x 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

= 5353 size 12{ { { size 8{5} } over { size 8{3} } } } {} x 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

= 5656 size 12{ { { size 8{5} } over { size 8{6} } } } {}m = 83, 3.3. size 12{ {3} cSup { size 8{ "." } } } {} cm

b) 5656 size 12{ { { size 8{5} } over { size 8{6} } } } {} x 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {} = 518518 size 12{ { { size 8{5} } over { size 8{"18"} } } } {} m

= 27, 7.7. size 12{ {7} cSup { size 8{ "." } } } {} cm

22.

(a) 32

(b) 15

(c) 25

(d) 25

(e) 45

(f) 2

(g) 8

(h) 7

(i) 7

(j) 6

(k) 6

(l) 8

(m) 8

(n) 8

(o) 100

Leerders Afdeling

Inhoud

AKTIWITEIT: Vermenigvuldiging van breuke [LU 1.7.3, LU 2.1.5]

18. VERMENIGVULDIGING VAN BREUKE

18.1 Vermenigvuldiging van breuke met natuurlike getalle

Jy weet reeds dat vermenigvuldiging eintlik herhaalde optelling is.

a) Kyk of jy die volgende tabel kan voltooi:

Figure 2
Figure 2 (graphics2.png)

b) Kyk goed na die voltooide tabel. Kan jy aan ’n korter manier / metode dink om die antwoorde te vind?

..........................................................................................................................................

..........................................................................................................................................

..........................................................................................................................................

c) LET OP!

Jy kan ook dié metode volg:

1. Skryf albei getalle as breuke, bv. 6×14=61×146×14=61×14 size 12{6 times { { size 8{1} } over { size 8{4} } } = { { size 8{6} } over { size 8{1} } } times { { size 8{1} } over { size 8{4} } } } {}

2. Vermenigvuldig die tellers met mekaar: 6 × 1 = 6

3. Vermenigvuldig die noemers met mekaar: 1 × 4 = 4

4. Vereenvoudig die antwoord: 64=124=11264=124=112 size 12{ { { size 8{6} } over { size 8{4} } } =1 { { size 8{2} } over { size 8{4} } } =1 { { size 8{1} } over { size 8{2} } } } {}

d) Bereken:

(i) 7×3107×310 size 12{7 times { { size 8{3} } over { size 8{"10"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(ii) 23×623×6 size 12{ { { size 8{2} } over { size 8{3} } } times 6} {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(iii) 12×7912×79 size 12{"12" times { { size 8{7} } over { size 8{9} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

e) Op ’n getallelyn sou ons 6×146×14 size 12{6 times { { size 8{1} } over { size 8{4} } } } {} so kon voorstel:

Figure 3
Figure 3 (graphics3.png)

f) Stel die volgende op ’n getallelyn voor: x=4×23x=4×23 size 12{x=4 times { { size 8{2} } over { size 8{3} } } } {}

18.2 Vermenigvuldiging van breuke met breuke

a) Kyk goed na die volgende voorbeelde:

(i) Die helfte ( 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}) van ’n driekwart ( 3434 size 12{ { { size 8{3} } over { size 8{4} } } } {}) kan so voorgestel word:

Figure 4
Figure 4 (graphics4.png)

Dus: 12×34=3812×34=38 size 12{ { { size 8{1} } over { size 8{2} } } times { { size 8{3} } over { size 8{4} } } = { { size 8{3} } over { size 8{8} } } } {}

(ii) Een derde ( 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {}) van ’n half ( 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}) lyk so:

Figure 5
Figure 5 (graphics5.png)

Dus 13×12=1613×12=16 size 12{ { { size 8{1} } over { size 8{3} } } times { { size 8{1} } over { size 8{2} } } = { { size 8{1} } over { size 8{6} } } } {}

b) Maak nou soortgelyke sketse vir:

(i) 15×1215×12 size 12{ { { size 8{1} } over { size 8{5} } } times { { size 8{1} } over { size 8{2} } } } {}

(ii) 310×12310×12 size 12{ { { size 8{3} } over { size 8{"10"} } } times { { size 8{1} } over { size 8{2} } } } {}

c) LET OP!

As ons ’n breuk met ’n breuk vermenigvuldig, bv. 2 3 × 3 8 2 3 × 3 8 size 12{ { { size 8{2} } over { size 8{3} } } times { { size 8{3} } over { size 8{8} } } } {}

1. Vermenigvuldig ons eers die tellers met mekaar: 2 × 3 = 6

2. Dan vermenigvuldig ons die noemers met mekaar: 3 × 8 = 24

3. Ons vereenvoudig ook waar nodig: 6 ÷ 6 24 ÷ 6 = 1 4 6 ÷ 6 24 ÷ 6 = 1 4 size 12{ { { size 8{6~ div ~6} } over { size 8{"24"~ div ~6} } } = { { size 8{1} } over { size 8{4} } } } {}

d) Onthou jy nog?

Om te kan vereenvoudig, moet jy altyd die teller en die noemer deur dieselfde getal deel.

e) Het jy geweet?

Ons kan ook van kansellering gebruik maak om die produk te bepaal.

Dit behels die deling van die teller EN die noemer deur ’n gemeenskaplike faktor

Figure 6
Figure 6 (graphics6.png)

Nog korter lyk dit so:

Figure 7
Figure 7 (graphics7.png)

Ons kan soos volg kanselleer: i) oorkruis ii) vertikaal

f) Bereken die volgende deur van kansellering gebruik te maak:

(i) k=158×45k=158×45 size 12{k= { { size 8{"15"} } over { size 8{8} } } times { { size 8{4} } over { size 8{5} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(ii) c=1825×4027c=1825×4027 size 12{c= { { size 8{"18"} } over { size 8{"25"} } } times { { size 8{"40"} } over { size 8{"27"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

18.3 Vermenigvuldiging van gemengde getalle

a) LET OP!

Ons moet eers die gemengde getal na ’n onegte breuk herlei en dan vermenigvuldig.

Bv. 6 × 1 3 4 = 6 1 × 7 4 42 4 10 2 4 10 1 2 6 × 1 3 4 = 6 1 × 7 4 42 4 10 2 4 10 1 2 alignl { stack { size 12{6 times 1 { { size 8{3} } over { size 8{4} } } = { { size 8{6} } over { size 8{1} } } times { { size 8{7} } over { size 8{4} } } } {} # = { { size 8{"42"} } over { size 8{4} } } {} # ="10" { { size 8{2} } over { size 8{4} } } {} # ="10" { { size 8{1} } over { size 8{2} } } {} } } {}

Onthou om altyd die antwoord te vereenvoudig!

b) Bereken die volgende en vereenvoudig waar moontlik:

(i) m=834×165m=834×165 size 12{m=8 { { size 8{3} } over { size 8{4} } } times { { size 8{"16"} } over { size 8{5} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(ii) n=523×910n=523×910 size 12{n=5 { { size 8{2} } over { size 8{3} } } times { { size 8{9} } over { size 8{"10"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

(iii) p=335×223p=335×223 size 12{p=3 { { size 8{3} } over { size 8{5} } } times 2 { { size 8{2} } over { size 8{3} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

19.1 Kan jy die antwoorde van die volgende ook bereken?

a) h=34×43h=34×43 size 12{h= { { size 8{3} } over { size 8{4} } } times { { size 8{4} } over { size 8{3} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

b) f=79×97f=79×97 size 12{f= { { size 8{7} } over { size 8{9} } } times { { size 8{9} } over { size 8{7} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

c) e=89×98e=89×98 size 12{e= { { size 8{8} } over { size 8{9} } } times { { size 8{9} } over { size 8{8} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

d) b=1215×1512b=1215×1512 size 12{b= { { size 8{"12"} } over { size 8{"15"} } } times { { size 8{"15"} } over { size 8{"12"} } } } {}

___________________________________________________

___________________________________________________

___________________________________________________

___________________________________________________

19.2 Wat merk jy op as jy na die antwoorde van bogenoemde kyk?

________________________________________________________________

________________________________________________________________

19.3 Het jy geweet?

As die produk van twee breuke 1 is, is die een breuk die resiprook van die ander. Om dus die resiprook van ’n breuk te kry, ruil ons net die teller en die noemer om!

19.4 Bepaal nou die resiprook van die volgende:

a) 17201720 size 12{ { { size 8{"17"} } over { size 8{"20"} } } } {} = ___________________________________________________

b) 40 = ___________________________________________________

c) 615615 size 12{6 { { size 8{1} } over { size 8{5} } } } {} = ___________________________________________________

d) 918918 size 12{9 { { size 8{1} } over { size 8{8} } } } {} = ___________________________________________________

19.5 Verduidelik aan ’n maat hoe jy die antwoorde van c en d hierbo gekry het. Skryf eers jou antwoord hier neer.

________________________________________________________________

________________________________________________________________

20. KOPKRAPPER!

Jodi is 123123 size 12{1 { { size 8{2} } over { size 8{3} } } } {} meter lank. Sandy is die helfte van haar lengte. Grace is een derde van Sandy se lengte.

a) Hoe lank is Sandy?

___________________________________________________

___________________________________________________

b) Hoe lank is Grace?

___________________________________________________

___________________________________________________

21. Tyd vir selfassessering!

Table 1
  • Merk die toepaslike blokkie:
 JA NEE
Ek kan breuke korrek optel    
Ek kan die kleinste gemene veelvoud van twee noemers bepaal.    
Ek kan breuke korrek aftrek    
Ek kan breuke met natuurlike getalle vermenigvuldig    
Ek kan breuke met breuke vermenigvuldig    
Ek kan breuke met gemengde getalle vermenigvuldig    
Ek weet hoe om van kansellering gebruik te maak as ek vermenigvuldig    
Ek weet hoe om te vereenvoudig    
Ek kan die resiprook van ’n breuk bepaal    

22. Kom ons kyk nou eers weer hoe flink jy nog kan dink! Kyk of jy die volgende hoofrekentoets binne 2 minute kan voltooi:

Table 2
   
a) 17 + 15 = ............ i) 110110 size 12{ { { size 8{1} } over { size 8{"10"} } } } {} van 70 = ............
b) 27 + ............ = 42 j) 110110 size 12{ { { size 8{1} } over { size 8{"10"} } } } {} van 1 minuut is ............ sekondes
c) 52 – 27 = ............ k) 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} van ’n dag = ............ uur
d) 72 – 47 = ............ l) 56 ÷ 7 = ............
e) 82 – 37 = ............ m) 72 ÷ 9 = ............
f) 10 × 1515 size 12{ { { size 8{1} } over { size 8{5} } } } {} = ............ n) 48 ÷ 6 = ............
g) 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} van 32 = ............ o) 104 ÷ 102 = ............
h) 1818 size 12{ { { size 8{1} } over { size 8{8} } } } {} van 56 = ............  

(15)

  • Kleur in:
Figure 8
Figure 8 (graphics8.png)

Assessering

Leeruitkomste 1: Die leerder is in staat om getalle en die verwantskappe daarvan te herken, te beskryf en voor te stel, en om tydens probleemoplossing bevoeg en met selfvertroue te tel, te skat, te bereken en te kontroleer.

Assesseringstandaard 1.7: Dit is duidelik wanneer die leerder skat en bereken deur geskikte bewerkings vir probleme wat die volgende behels, kies en gebruik:

1.7.3: optelling, aftrekking en vermenigvuldiging van gewone breuke.

Leeruitkomste 2: Die leerder is in staat om patrone en verwantskappe te herken, te beskryf en voor te stel en probleme op te los deur algebraïese taal en vaardighede te gebruik.

Assesseringstandaard 2.1: Dit is duidelik wanneer die leerder numeriese en meetkundige patrone ondersoek en uitbrei op soek na ‘n verwantskap of reëls, insluitend patrone;

2.1.5: voorgestel in tabelle.

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