Fractions - Fractions - R R evision evision
Mar 28, 2015
Fractions - Fractions - RRevisionevision
Fractions are numbers which mostly descFractions are numbers which mostly descrribe ______________.ibe ______________.parts of parts of a wholea whole
E.g.E.g. colored:colored: ______551212
of of the the rectanglerectangle
not colored:not colored: ______77
1212of of the the rectanglerectangle
a)a) colored:colored: ______7733
circlescircles
not colored:not colored: ______22
33of of a a circlecircle
b)b)colored :colored :
not colored:not colored:
____3355
rectanglesrectangles22
______22
55of of a a rectanglerectangle
Fractions can descFractions can descrribe even more than one whole!ibe even more than one whole!
E.g.E.g.
How to color:How to color:
a)a) ______5566
of of a a parallelogramparallelogram??
b)b) ______7744
squaressquares??
c)c) ______1133
rhombusesrhombuses??44
The parts of the fraction:The parts of the fraction:
____aa
bb
?? numeratornumerator
?? denominatordenominator
?? fraction linefraction line (or (or vinculum)vinculum)
Denominator Denominator tells ustells usinintoto how many how many equalequal parts parts the wholethe whole
Numerator Numerator tells tells usus
how many how many of those of those parts should be colored.parts should be colored.
Fraction line always meansFraction line always meansdivision.division.
E.g.E.g. ____88
44== 8 : 4 =8 : 4 = 22
We used these properties of numerator and We used these properties of numerator and denominator in thedenominator in the previous examples.previous examples.
==
should beshould be divided divided..
ProperProper fractions are fractions with fractions are fractions withthe the numerator less thannumerator less than
ImproperImproper fractions are fractions with fractions are fractions withtthe he numerator greaternumerator greater
than than or or equal to equal to the the denominator.denominator.
E.g. E.g. , , ______1144
, , ______2299
______991010
... are _____ fractions. ... are _____ fractions.
They are ______ than 1.They are ______ than 1.lessless
<<
properproper
the the denominator.denominator.
improperimproper
They are __________________ 1.They are __________________ 1.greater than or equal togreater than or equal to
E.g. E.g. , , ______9944
, , ______5533
______4444
... are ________ fractions. ... are ________ fractions. , , ______6622
≥≥
Which of Which of the the fractions above are equal to 1?fractions above are equal to 1?
How can we recognize fractions which are equal to 1?How can we recognize fractions which are equal to 1?
TheThe numerator numerator is is equal to equal to the the denominator!!denominator!!
ProperProper fractions are fractions with fractions are fractions withthe the numerator less thannumerator less than
ImproperImproper fractions are fractions with fractions are fractions withthe the numerator greaternumerator greater
than or than or equal to the denominator.equal to the denominator.
the the denominator.denominator.
≥≥
Which of Which of the the fractions above are greater than 1?fractions above are greater than 1?
How can we recognize fractions which are greater than 1?How can we recognize fractions which are greater than 1?
The numerator The numerator is is greater than greater than the the denominator!!denominator!!
ImproperImproper fractions are fractions with fractions are fractions withthe the numerator greaternumerator greater
than or than or equal to equal to the the denominator.denominator.
improperimproperE.g. E.g. , , ______9944
, , ______5533
______4444
... are ________ fractions. ... are ________ fractions. , , ______6622
They are __________________ 1.They are __________________ 1.greater than or equal togreater than or equal to
≥≥
Any Any improperimproper fraction can be changed into fraction can be changed intoa mixed fraction a mixed fraction orora a natural number.natural number.
ImproperImproper fractions are fractions with fractions are fractions withthe the numerator greaternumerator greater
than or than or equal to equal to the the denominator.denominator.
improperimproperE.g. E.g. , , ______9944
, , ______5533
______4444
... are ________ fractions. ... are ________ fractions. , , ______6622
They are __________________ 1.They are __________________ 1.greater than or equal togreater than or equal to
≥≥
Which of Which of these these fractions can be changed into fractions can be changed into mixed fractions mixed fractions ??
Change them (look at the picture)!Change them (look at the picture)!
______9944
== 22 ______1144
______5533
== 11 ______2233
ImproperImproper fractions are fractions with fractions are fractions withthe the numerator greaternumerator greater
than or than or equal to equal to the the denominator.denominator.
improperimproperE.g. E.g. , , ______9944
, , ______5533
______4444
... are ________ fractions. ... are ________ fractions. , , ______6622
≥≥
Which of Which of these these fractions can be changed into fractions can be changed into natural numbers natural numbers ??
______4444
== 11 ______6622
== 33
ImproperImproper fractions are fractions with fractions are fractions withthe the numerator greater numerator greater
than or than or equal to equal to the the denominator.denominator.
improperimproperE.g. E.g. , , ______9944
, , ______5533
______4444
... are ________ fractions. ... are ________ fractions. , , ______6622
Change them (look at the picture)!Change them (look at the picture)!
Now, let's revise how to Now, let's revise how to calculatecalculate it (without it (without a a picture)!picture)!
1.) Change into 1.) Change into a a mixed mixed numbernumber or or aa natural number natural number (do what is possible)(do what is possible)::
______191988
==a)a) 3388
______22
Explanation:Explanation:19:8 equals 19:8 equals 22 and remainder and remainder 33
Rewrite denominator!Rewrite denominator!
==
Now, let's revise how to Now, let's revise how to calculatecalculate it (without it (without a a picture)!picture)!
1.) Change into 1.) Change into a a mixed mixed numbernumber or or aa natural number natural number (do what is possible)(do what is possible)::
______191988
==a)a) 3388
______22
______686899
==b)b) 5599
______77
______424277
==c)c) 66
Explanation:Explanation:42:7 equals 42:7 equals 66 (no remainder) (no remainder)
==
Now, let's revise how to Now, let's revise how to calculatecalculate it (without it (without a a picture)!picture)!
1.) Change into 1.) Change into a a mixed mixed numbernumber or or aa natural number natural number (do what is possible)(do what is possible)::
______191988
==a)a) 3388
______22
______686899
==b)b) 5599
______77
______424277
==c)c) 66
______363644
==d)d)
______2299
==e)e) This is This is properproper fraction fraction(numerator is less than denominator),(numerator is less than denominator),so we so we can'tcan't change it into change it into a a mixed mixed fraction or into natural number!fraction or into natural number!
99
How did we calculate How did we calculate in all these tasks?in all these tasks?
We divided We divided the the numeratornumeratorby by the the denominator.denominator.
Why?Why?
Because the fraction lineBecause the fraction linealwaysalways means division!means division!
Now let's change a fraction Now let's change a fraction into a into a decimaldecimal number! number!
How to do it?How to do it?
We should divide again,We should divide again,but now but now in writingin writing......
Let's revise it...Let's revise it...
Let's change the number at task "a)" into a Let's change the number at task "a)" into a decimaldecimal number... number...How to do it?How to do it?
______191988
== 19 : 8 =19 : 8 = 22
3300..33
6600
77
4400
55
00
So, we changed the same fractionSo, we changed the same fractioninto both - mixed fraction andinto both - mixed fraction and
decimal number.decimal number.
Remember:Remember:
When we change fractionWhen we change fractioninto any another form,into any another form,
we we always dividealways divide(because the fraction line(because the fraction line
means division)!means division)!
Only when we change into Only when we change into decimaldecimal number, number,
then we then we use use long divisionlong division..
______191988
==a)a) ______223388
Now, let's revise how to Now, let's revise how to calculatecalculate it (without it (without a a picture)!picture)!
1.) Change into 1.) Change into a a mixed mixed numbernumber or or aa natural number natural number (do what is possible)(do what is possible)::
Now conversely! Let's revise how to change numbers Now conversely! Let's revise how to change numbers from other forms into fractions...from other forms into fractions...
2.) Change into fraction:2.) Change into fraction:
885555____________77
88==a)a) 66
6 6 ·· 8 + 7 8 + 7
RewriteRewritedenominator!denominator!
==
Now conversely! Let's revise how to change numbers Now conversely! Let's revise how to change numbers from other forms into fractions...from other forms into fractions...
2.) Change into fraction:2.) Change into fraction:
885555____________77
88==a)a) 66
776969____________66
77==b)b) 99
c) 8 =c) 8 =1188______ ==
1616______22
==2424______33
= ...= ...
(When we divide numerator by denominator, the result must be (When we divide numerator by denominator, the result must be 88 !) !)
== == == == == == = ...= ...
Now conversely! Let's revise how to change numbers Now conversely! Let's revise how to change numbers from other forms into fractions...from other forms into fractions...
2.) Change into fraction:2.) Change into fraction:
885555____________77
88==a)a) 66
776969____________66
77==b)b) 99
c) 8 =c) 8 =1188______ ==
1616______22
==2424______33
= ...= ...
d) 2.41 =d) 2.41 =100100241241________ Rewrite the given number, but without decimal Rewrite the given number, but without decimal pointpoint......
Write the digit 1 and as many zeros as we have Write the digit 1 and as many zeros as we have decimal digits in the given number...decimal digits in the given number...2 decimal digits2 decimal digits
2 zeros2 zeros
Explanation:Explanation:
==
Now conversely! Let's revise how to change numbers Now conversely! Let's revise how to change numbers from other forms into fractions...from other forms into fractions...
2.) Change into fraction:2.) Change into fraction:
885555____________77
88==a)a) 66
776969____________66
77==b)b) 99
c) 8 =c) 8 =1188______ ==
1616______22
==2424______33
= ...= ...
d) 2.41 =d) 2.41 =100100241241________
e) 30.9 =e) 30.9 =1010309309________
f) 0.019 =f) 0.019 =100010001919________
773131____________33
77==h)h) 44
g) 27 =g) 27 =112727______ ==
5454______22
= ...= ...
i) 28.93 =i) 28.93 =10010028932893________
Some decimal numbers can be changed into mixed fractions.Some decimal numbers can be changed into mixed fractions.Let's revise it...Let's revise it...
3.) Change into mixed number:3.) Change into mixed number:
a) 2.41 =a) 2.41 =1001004141________22
100100
241241______
Recall:Recall:2.41 can be changed not only into 2.41 can be changed not only into a a mixed fraction, but intomixed fraction, but into an an improper improper fraction as well.fraction as well.Say that improper fraction...Say that improper fraction...
2 decimal digits2 decimal digits2 zeros2 zeros
==
Some decimal numbers can be changed into mixed fractions.Some decimal numbers can be changed into mixed fractions.Let's revise it...Let's revise it...
3.) Change into mixed number:3.) Change into mixed number:
a) 2.41 =a) 2.41 =1001004141________22
b) 30.9 =b) 30.9 =101099________3030
c) 15.007 =c) 15.007 =10001000
77________1515
d) 0.045 =d) 0.045 =This decimal numberThis decimal numbercan'tcan't be changed into be changed into a a mixed fractionmixed fractionbecause it has got because it has got zero wholeszero wholes..We can We can only only change it into change it into a a fraction.fraction.
If we should change it into fraction,If we should change it into fraction,
the solution would be the solution would be 100010004545________ ..
What does it mean - "to reduce aWhat does it mean - "to reduce a fraction" ?fraction" ?
To reduce aTo reduce a fraction means to divide both fraction means to divide both the the numerator and numerator and ddenominator of the fraction by the enominator of the fraction by the samesame number.number.
When we reduce a fraction, does When we reduce a fraction, does the the reduced fraction represent a reduced fraction represent a bigger or a smaller part than the bigger or a smaller part than the originaloriginal fraction? fraction?
They represent equal parts!!They represent equal parts!!
4.) Reduce these fractions to non-reducible fractions:4.) Reduce these fractions to non-reducible fractions:
______10101212
==a)a) We can reduce it by __.22
55
66
______5566
==
When we reduce a fraction to the non-reducible fraction, When we reduce a fraction to the non-reducible fraction, then we get this fraction in the then we get this fraction in the lowest possible termslowest possible terms..
So, we divide both the numerator and denominator by 2 and write the results…
What does it mean - "to reduce aWhat does it mean - "to reduce a fraction" ?fraction" ?
To reduce aTo reduce a fraction means to divide both the numerator and fraction means to divide both the numerator and ddenominator of the fraction by the enominator of the fraction by the samesame number.number.
They represent equal parts!!They represent equal parts!!
4.) Reduce these fractions to non-reducible fractions:4.) Reduce these fractions to non-reducible fractions:
______24243030
==b)b) We can reduce it by __.66
44
55
______4455
==
When we reduce a fraction, does When we reduce a fraction, does the the reduced fraction represent a reduced fraction represent a bigger or a smaller part than the bigger or a smaller part than the originaloriginal fraction? fraction?
When we reduce a fraction to the non-reducible fraction, When we reduce a fraction to the non-reducible fraction, then we get this fraction in the then we get this fraction in the lowest possible termslowest possible terms..
What does it mean - "to reduce aWhat does it mean - "to reduce a fraction" ?fraction" ?
To reduce aTo reduce a fraction means to divide both fraction means to divide both the the numerator and numerator and ddenominator of the fraction by the enominator of the fraction by the samesame number.number.
They represent equal parts!!They represent equal parts!!
4.) Reduce these fractions to non-reducible fractions:4.) Reduce these fractions to non-reducible fractions:
______42426363
==c)c) We can reduce it by __.77
66
99
______6699
Now we can reduce again, by __.33
22
33
______2233
==
== ==
When we reduce a fraction, does When we reduce a fraction, does the the reduced fraction represent a reduced fraction represent a bigger or a smaller part than the bigger or a smaller part than the originaloriginal fraction? fraction?
When we reduce a fraction to the non-reducible fraction, When we reduce a fraction to the non-reducible fraction, then we get this fraction in the then we get this fraction in the lowest possible termslowest possible terms..
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
a) One whole a) One whole equalsequals ____ fourths. ____ fourths.fourfour 1 =1 = ______4444
==
b) One whole b) One whole equalsequals _____ twelfths. _____ twelfths.twelvetwelve 1 =1 = ______12121212
==
c) Two wholes c) Two wholes equalequal ___ thirds. ___ thirds.sixsix 2 =2 = ______6633
==
5.) Complete these sentences:5.) Complete these sentences:
d) 4 days = weekd) 4 days = week______4477
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
5.) Complete:5.) Complete:
Explanation:Explanation:
First, let's recall that a week has ___ days.First, let's recall that a week has ___ days.77
So we can conclude that each day is of the week.So we can conclude that each day is of the week.____1177
Now, we should think in this way:Now, we should think in this way:
If If 1 day1 day is is of the weekof the week,,____1177
then then 2 days2 days are are of the weekof the week,,____2277
3 days3 days are are of the weekof the week,,____3377
and and 4 days4 days are are of the weekof the week..____4477
d) 4 days = weekd) 4 days = week______4477
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
5.) Complete:5.) Complete:
We can explain it in another way:We can explain it in another way:
Again, let's recall that a week has ___ days.Again, let's recall that a week has ___ days.77
MondayTuesday
WednesdayThursday
FridaySaturdaySunday
week:week:
So, let's consider it together with the rectangle divided into So, let's consider it together with the rectangle divided into 77 equal parts!equal parts!
rectangle:rectangle:
Left picture:Left picture:We are interested in We are interested in which part of the week consist of which part of the week consist of 4 4 daysdays......
Right picture:Right picture:Which part of the Which part of the rectangle consist of rectangle consist of 4 4 partsparts......
??____4477
of the weekof the week
d) 4 days = weekd) 4 days = week______4477
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
5.) Complete:5.) Complete:
Shortly:Shortly:
Rewrite the given number into numerator.Rewrite the given number into numerator.
In the denominator write the In the denominator write the total numbertotal number of days in a week ! of days in a week !
As we already know,As we already know,
the the denominatordenominator is the is the total numbertotal number of equal parts, of equal parts,
and the and the numerator numerator describes the number of parts we are describes the number of parts we are interested ininterested in..
d) 4 days = weekd) 4 days = week______4477
e) 20 min. = houre) 20 min. = hour______20206060
= hour= hour______1133
We can reduce it by __.2020
11
33
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
5.) Complete:5.) Complete:
f) 5 months = yearf) 5 months = year______551212
12
39
6
____1133
min.min.??2020
hourhour??
JanuaryFebruary
MarchAprilMayJuneJuly
year:year:
AugustSeptember
OctobarNovemberDecember
____551212
of the of the yearyear
??
6.) There are 7 tasks in the Ivan's homework.6.) There are 7 tasks in the Ivan's homework.
Ivan solved 2 of them.Ivan solved 2 of them.
What portionWhat portion of the homework did Ivan solve? of the homework did Ivan solve?
Ivan solved of his homework.Ivan solved of his homework.______2277
What portionWhat portion of the homework does he still have to solve? of the homework does he still have to solve?
He should solveHe should solve of his homework yet. of his homework yet.______5577
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
1st task2nd task3rd task4th task5th task6th task7th task
homework:homework:
??____2277
of the homeworkof the homework
????________5555
7777of the homeworkof the homeworkof the homeworkof the homework
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
If children ate If children ate (seven tenths) of (seven tenths) of a a cake, whichcake, which
fraction fraction of of the the cake remained?cake remained?
______771010
7.) 7.)
______331010
(three tenths) of (three tenths) of the the cakecake remained remained..
______331010
of the cakeof the cake ______771010
of the cakeof the cake
8.) Climber Dario climbed of his path in one hour,8.) Climber Dario climbed of his path in one hour,______661414
another of the path in the next one hour, another of the path in the next one hour,
and finally of his path in the third hour.and finally of his path in the third hour.
Did he climb his whole path?Did he climb his whole path?
______441414
______331414
______13131414
______441414
______661414
++ ______331414
++ ==
No, he didn't climb his whole path.No, he didn't climb his whole path.
There remained of his path.There remained of his path.______111414
Now let's revise other properties of fractions...Now let's revise other properties of fractions...
9.) Seven friends gathered some money and bought 3 chocolates9.) Seven friends gathered some money and bought 3 chocolates
ofof equal equal sizesize. They want to divide . They want to divide the the chocolates equally.chocolates equally. How How muchmuch chocolate will each of them get? chocolate will each of them get?
3 : 7 =3 : 7 = ______3377
Each friend will get of Each friend will get of a a chocolate.chocolate.______3377
Explanation with pictures:Explanation with pictures:
When they divide the first chocolate into 7 equal parts,When they divide the first chocolate into 7 equal parts,
each friend will get each friend will get ____11
77of of the the chocolate.chocolate.
When they divide the second chocolate into 7 equal parts,When they divide the second chocolate into 7 equal parts,
each friend will get each friend will get ____11
77of of the the chocolate.chocolate.
When they divide the third chocolate into 7 equal parts,When they divide the third chocolate into 7 equal parts,
each friend will get each friend will get ____11
77of of the the chocolate.chocolate.
So, after all divisions each friend will haveSo, after all divisions each friend will have____33
77of of a a chocolate.chocolate.
10.) a) 12 chocolates should be divided among 5 friends.10.) a) 12 chocolates should be divided among 5 friends. How many chocolates will each of them get?How many chocolates will each of them get?
12 : 5 =12 : 5 = ______121255
==2255
______22
Each friend will get 2 chocolates.Each friend will get 2 chocolates.______2255
Explanation with pictures:Explanation with pictures:
Each friend getsEach friend gets____11
55of the 11th chocolate.of the 11th chocolate.
Each friend getsEach friend gets____11
55of the 12th chocolate.of the 12th chocolate.
So, after all divisions each friend will haveSo, after all divisions each friend will have ____22
55chocolates!chocolates!22
10.) b) What about dividing 12 chocolates among 3 friends?10.) b) What about dividing 12 chocolates among 3 friends?
12 : 3 =12 : 3 = 44
Each friend will get 4 chocolates.Each friend will get 4 chocolates.
Explanation with pictures:Explanation with pictures:
After division each friend will haveAfter division each friend will have44 chocolates. chocolates.
11.) Little Ana ate strawberries. How many strawberries11.) Little Ana ate strawberries. How many strawberries
did she eat actually?did she eat actually?
______121222
12 : 2 =12 : 2 ===______121222
Little Ana ate 6 strawberries.Little Ana ate 6 strawberries.
66
Explanation with pictures:Explanation with pictures:
strawberriesstrawberries______1212
22= 6 strawberries= 6 strawberries
12.) There are 48 apples in the box.12.) There are 48 apples in the box.
______3388
of the box are red apples,of the box are red apples, ______551212
are green applesare green apples
and and the rest of the the rest of the apples are yellow. apples are yellow.
a) How many apples are of which color?a) How many apples are of which color?
______3388
of 48 isof 48 is 1818red:red:
______551212
of 48 isof 48 is 2020green:green:
( we calculated 48:8( we calculated 48:8··3 )3 )
18 + 20 =18 + 20 = 38, 38, yellow:yellow: 48 - 38 =48 - 38 = 10 10
There are 18 red, 20 green and 10 yellow apples in that box.There are 18 red, 20 green and 10 yellow apples in that box.
b) b) What portion What portion of the box do yellow apples form?of the box do yellow apples form?
______10104848
55
2424We can reduce it by __.22== ______55
2424Yellow apples form (five twenty-Yellow apples form (five twenty-fourths) of the box.fourths) of the box.
______552424
13.) Complete these expressions:13.) Complete these expressions:
a) of a) of 15 15 is is ______2255
66
We calculated:We calculated:
1515 : 5: 5 ·· 2 2 == 66
How to explain this calculation?How to explain this calculation?
we divide it into we divide it into 55 equal parts, and then color equal parts, and then color 22 parts. parts.
Recall: If we want to color of some figure, thenRecall: If we want to color of some figure, then______2255
Here we do just about the sameHere we do just about the same thing! thing!
number 15number 15 into into 55 equal parts, and then take equal parts, and then take 22 parts. parts.
of 15 can be calculated so that we divideof 15 can be calculated so that we divide______2255
1515 : 5: 5 ·· 2 2 == 66
a) of a) of 15 15 is is ______2255
13.) Complete these expressions:13.) Complete these expressions:
66
b) of b) of 72 72 is is ______4499
3232
c) of c) of 10 10 is is ______3344
Here we can't divide 10:4, so we must calculate inHere we can't divide 10:4, so we must calculate insome another way!some another way!
The word The word ofof means meansmultiplication!multiplication!______3344
·· 101055
22
We can reduce it by __.22
______151522
== 77 ______1122
77 ______1122
a) of a) of 15 15 is is ______2255
13.) Complete these expressions:13.) Complete these expressions:
Are weAre we allowed to calculate in that way in the allowed to calculate in that way in thea a and and bb tasks? tasks?
Yes, we are!!!Yes, we are!!!
==
E.g.E.g. ______2255
·· 151533
11
We can reduce it by __.55
______6611
== 66==
Both proceduresBoth proceduresgive equal results!!!give equal results!!!
66
5 61 2 3 4 7 8 9 10
b) of b) of 72 72 is is ______4499
3232
c) of c) of 10 10 is is ______3344
77 ______1122
a) of a) of 15 15 is is ______2255
13.) Complete these expressions:13.) Complete these expressions:
How How can wecan we imagineimagine the problem the problem in part cin part c??
66
We have 10 pieces of something, e.g. 10 pears...We have 10 pieces of something, e.g. 10 pears...The dThe denominator 4 tells us that we should divide this "bulk" enominator 4 tells us that we should divide this "bulk" into 4 equal parts... into 4 equal parts...
1st1st 2nd2nd 3rd3rd 4th4th
The nThe numerator 3 tells us to take 3 of these 4 parts...umerator 3 tells us to take 3 of these 4 parts...
How manHow manyy pears do we have in these three parts? pears do we have in these three parts? ____112277
We got the same result !!!We got the same result !!!
d) of d) of is is ______12121313
______39394444
______12121313
·· ______39394444
==33
1111
We can reduce it by __.44
11
33
We can reduce it by __.1313
______991111
______991111
a) of a) of 15 15 is is ______2255
13.) Complete these expressions:13.) Complete these expressions:
b) of b) of 72 72 is is ______4499
3232
c) of c) of 10 10 is is ______3344
77 ______1122
66
14.) Complete:14.) Complete:
a) is ______ than 1a) is ______ than 1______2255
lessless , by , by ______3355
..
Explanation with pictures:Explanation with pictures:
of the rectangleof the rectangle is less than is less than 1 rectangle1 rectangle,,______22
55
<<
by the by the uncoloreduncolored part, and this part is part, and this part is______33
55of the rectangle.of the rectangle.
14.) Complete:14.) Complete:
a) is ______ than 1a) is ______ than 1______2255
lessless , by , by ______3355
..
b) is _______ than 1b) is _______ than 1______19191515
greatergreater , by, by ______441515
..
Explanation with pictures:Explanation with pictures:
rectanglesrectangles is greater than is greater than 1 rectangle1 rectangle,,______1919
1515
>>
by the part determined by by the part determined by secondsecond rectangle, and it is rectangle, and it is
______44
1515of the rectangleof the rectangle..
15.) 15.) In each inequality below, whichIn each inequality below, which number is number is the the greatergreater??
a)a) ______8899
______5599
>>
>>
14.) Complete:14.) Complete:
a) is ______ than 1a) is ______ than 1______2255
lessless , by , by ______3355
..
b) is _______ than 1b) is _______ than 1______19191515
greatergreater , by, by ______441515
..
15.) 15.) In each inequality below, whichIn each inequality below, which number is number is the the greatergreater??
a)a) ______8899
______5599
>>
<<b)b) ______771010
______1155
22 44
<<
14.) Complete:14.) Complete:
a) is ______ than 1a) is ______ than 1______2255
lessless , by , by ______3355
..
b) is _______ than 1b) is _______ than 1______19191515
greatergreater , by, by ______441515
..
15.) 15.) In each inequality below, whichIn each inequality below, which number is number is the the greatergreater??
a)a) ______8899
______5599
>>
<<b)b) ______771010
______1155
22 44
<<c)c) ______331111
44 55
<<
14.) Complete:14.) Complete:
a) is ______ than 1a) is ______ than 1______2255
lessless , by , by ______3355
..
b) is _______ than 1b) is _______ than 1______19191515
greatergreater , by, by ______441515
..
15.) 15.) In each inequality below, whichIn each inequality below, which number is number is the the greatergreater??
a)a) ______8899
______5599
>>
<<b)b) ______771010
______1155
22 44
<<c)c) ______331111
44 55
d)d) ______3355
______1155
66 66>>
e)e) ______8833
______5522
We multiply through diagonals...We multiply through diagonals...
·
1616
·
1515>>
>>
Instead of Instead of cross-cross-multiplying,multiplying,we can find the common denominatorwe can find the common denominatorand then compare numerators...and then compare numerators...
If we would take the common If we would take the common denominator denominator 33··22, that is 6,, that is 6,then we would get numeratorsthen we would get numerators1616 and and 1515. Multiplication through. Multiplication throughdiagonals is shortcut for that.diagonals is shortcut for that.>>
14.) Complete:14.) Complete:
a) is ______ than 1a) is ______ than 1______2255
lessless , by , by ______3355
..
b) is _______ than 1b) is _______ than 1______19191515
greatergreater , by, by ______441515
..
15.) 15.) In each inequality below, whichIn each inequality below, which number is number is the the greatergreater??
a)a) ______8899
______5599
>>
<<b)b) ______771010
______1155
22 44
<<c)c) ______331111
44 55
d)d) ______3355
______1155
66 66>>
e)e) ______8833
______5522>>
>>
14.) Complete:14.) Complete:
a) is ______ than 1a) is ______ than 1______2255
lessless , by , by ______3355
..
b) is _______ than 1b) is _______ than 1______19191515
greatergreater , by, by ______441515
..
16.) Complete:16.) Complete:
a) If the a) If the numeratornumerator increases, then the fraction _____________. increases, then the fraction _____________.increasesincreases as well as well
E.g.E.g. ______1133
______2233
______3333
______4433
______5533
If we are looking from the left to the right,If we are looking from the left to the right,the numerators ________.the numerators ________.increaseincrease
Colored parts, that is the fractions _____________.Colored parts, that is the fractions _____________.increase as wellincrease as well
b) If the b) If the denominatordenominator increases, then the fraction _________. increases, then the fraction _________.decreasesdecreases
E.g.E.g. ______1111
______1122
______1133
______1144
______1155
If we are looking from the left to the right,If we are looking from the left to the right,the denominators ________.the denominators ________.increaseincrease
Colored parts, that isColored parts, that is,, the fractions the fractions of the whole of the whole _______._______.decreasedecrease
16.) Complete:16.) Complete:
a) If the a) If the numeratornumerator increases, then the fraction _____________. increases, then the fraction _____________.increasesincreases as well as well
16.) Complete:16.) Complete:
a) If the a) If the numeratornumerator increases, then the fraction _____________. increases, then the fraction _____________.
b) If the b) If the denominatordenominator increases, then the fraction _________. increases, then the fraction _________.decreasesdecreases
increasesincreases as well as well
17.) Figure out:17.) Figure out:
______5566
++a)a) ______4499
== ________________15151818
==++ 88 ______2323
1818== 11 ______55
1818
________________272766
==-- 11 ______2626
66==
44 ______2266
______1122
--b)b) ______1166
==44 ______9922
-- ______1166
==
== We can reduce it by __.22
11
33
44 ______1133
==
18.) Try to figure 18.) Try to figure these these out out mentallymentally!!
5533 ______
77++a)a) ______55
77==33
Explanation with pictures:Explanation with pictures:
18.) Try to figure 18.) Try to figure these these out out mentallymentally!!
5533 ______
77++a)a) ______55
77==33
22______55
--b)b) ______3355
==11
Explanation with pictures:Explanation with pictures:
18.) Try to figure 18.) Try to figure these these out out mentallymentally!!
5533 ______
77++a)a) ______55
77==33
22______55
--b)b) ______3355
==11
7755 ______
99--c)c) ______22
99==66
Explanation with pictures:Explanation with pictures:
18.) Try to figure 18.) Try to figure these these out out mentallymentally!!
5533 ______
77++a)a) ______55
77==33
22______55
--b)b) ______3355
==11
7755 ______
99--c)c) ______22
99==66
______1122
++d)d) ==44 5511
99 ______22
Explanation with pictures:Explanation with pictures:
18.) Try to figure 18.) Try to figure these these out out mentallymentally!!
5533 ______
77++a)a) ______55
77==33
22______55
--b)b) ______3355
==11
7755 ______
99--c)c) ______22
99==66
______1122
++d)d) ==44 5511
99 ______22
______221111
--e)e) ==66 6622______
1111
Explanation with pictures:Explanation with pictures:
18.) Try to figure 18.) Try to figure these these out out mentallymentally!!
5533 ______
77++a)a) ______55
77==33
22______55
--b)b) ______3355
==11
7755 ______
99--c)c) ______22
99==66
______1122
++d)d) ==44 5511
99 ______22
______221111
--e)e) ==66 6622______
1111
______1133
--f)f) ______1133
==88 88
Explanation with pictures:Explanation with pictures:
18.) Try to figure 18.) Try to figure these these out out mentallymentally!!
5533 ______
77++a)a) ______55
77==33
22______55
--b)b) ______3355
==11
7755 ______
99--c)c) ______22
99==66
______1122
++d)d) ==44 5511
99 ______22
______221111
--e)e) ==66 6622______
1111
______1133
--f)f) ______1133
==88 88
--g)g) ______7788
==66 2211
33 ______88
Explanation with pictures:Explanation with pictures:
19.) Figure out:19.) Figure out:
______27274949
··a)a) ______212199
==33
11
We can reduce it by __.99
77
33
We can reduce it by __.77
______9977
== 11 ______2277
______2299
··b)b) ______54541919
==44 ______383899
·· ______54541919
==22
11
We can reduce it by __.1919
11
66
We can reduce it by __.99
______121211
== 1212
··c)c) ______1166
==99 22 ______9911
·· ______131366
33
22
We can reduce it by __.33
== ______393922
== 1919 ______1122
20.) Figure out:20.) Figure out:
::a)a) ______121277
==88 ______8811
·· ______771212
22
33
We can reduce it by __.44
== ______141433
== 44 ______2233
::b)b) ______7788
==33______1188
______1188
:: ______313188
== ______1188
·· ______883131
11
11
We can reduce it by __.88
== ______113131
Let's recall theLet's recall the “ “number linenumber line””....
00 11 22 33 44
Place the following numbers onto the number line:Place the following numbers onto the number line:
a)a) 22 ______5566
between __ and __between __ and __22 33
00 11 22 33 44
a)a) 22 ______5566
marked part of the number linemarked part of the number lineshould be divided into ____________should be divided into ____________6 equal parts6 equal parts
Place the following numbers onto the number line:Place the following numbers onto the number line:
Let's recall theLet's recall the “ “number linenumber line””....
00 11 22 33 44
a)a) 22 ______5566
we should count __ parts of the whole from the leftwe should count __ parts of the whole from the left55
22 ____55
66
Place the following numbers onto the number line:Place the following numbers onto the number line:
Let's recall theLet's recall the “ “number linenumber line””....
00 11 22 33 44
a)a) 22 ______5566
22 ____55
66
b)b) ______7722
== 33 ______1122
between __ and __between __ and __33 44 , exactly in the ______, exactly in the ______middlemiddle
33 ____11
22
____77
22
Place the following numbers onto the number line:Place the following numbers onto the number line:
Let's recall theLet's recall the “ “number linenumber line””....
00 11 22 33 44
a)a) 22 ______5566
22 ____55
66
b)b) ______7722
== 33 ______1122
33 ____11
22
____77
22
c)c) ______2233
it is not possible to change it into ait is not possible to change it into a mixed number, mixed number, there are no wholes,there are no wholes,so this number lies between __ and __so this number lies between __ and __00 11
Place the following numbers onto the number line:Place the following numbers onto the number line:
Let's recall theLet's recall the “ “number linenumber line””....
00 11 22 33 44
a)a) 22 ______5566
22 ____55
66
b)b) ______7722
== 33 ______1122
33 ____11
22
____77
22
c)c) ______2233 marked part of the number linemarked part of the number line
should be divided into ____________should be divided into ____________3 equal parts3 equal parts
Place the following numbers onto the number line:Place the following numbers onto the number line:
Let's recall theLet's recall the “ “number linenumber line””....
00 11 22 33 44
a)a) 22 ______5566
22 ____55
66
b)b) ______7722
== 33 ______1122
33 ____11
22
____77
22
c)c) ______2233
we count __ parts we count __ parts ofof the whole from the left the whole from the left22
____22
33
Place the following numbers onto the number line:Place the following numbers onto the number line:
Let's recall theLet's recall the “ “number linenumber line””....
We We shallshall continue this revision in writing continue this revision in writing......
Now we should Now we should be able to be able to solve several more complex tasks solve several more complex tasks with more with more ‘fraction ‘fraction calculatcalculation’ion’ operations. operations.
Open your notebooks...Open your notebooks...
Author of presentation:Author of presentation:
Antonija HorvatekAntonija HorvatekCroatiaCroatia , October 2008., October 2008.
With thanks to:With thanks to:
GSCGSCfor support, great for support, great sugsugggestions and estions and
preliminary helppreliminary help with with the the translation translation
ininto Englishto English
andand
Rex BoggsRex Boggsfor support and help for support and help
with with the the translationtranslation into into fluent U.S. idiom fluent U.S. idiom
(a.k.a. ‘American’)(a.k.a. ‘American’)..
You are welcome to use this presentation in your teaching.Additionally, you can change some parts of it if used solely for teaching.
However, if you want to use it in public lectures, workshops, in writing books, articles, on CDs or any public forum or for any commercial purpose, please ask for specific permission from the author.
Antonija Horvatekhttp://public.carnet.hr/[email protected]