Transcript Fractions - Revision

Fractions - Revision
parts of a whole
Fractions are numbers which mostly describe ______________.
E.g.
colored:
5
___
of the rectangle
12
7
___
of the rectangle
not colored:
12
Fractions can describe even more than one whole!
E.g.
a)
colored:
7
___
circles
3
2
not colored: ___ of a circle
3
b)
colored :
3
5
2 __ rectangles
2
not colored: ___ of a rectangle
5
How to color:
a)
5
___
of a parallelogram?
6
b)
7
___
squares?
4
c)
1
___
rhombuses?
4
3
The parts of the fraction:
?numerator
a
__
b
?fraction line (or vinculum)
?denominator
Denominator tells us into how many equal parts the whole
should be divided.
Numerator tells us how many of those parts should be colored.
We used these properties of numerator and denominator
in the previous examples.
Fraction line always means division.
E.g.
8
__
= 8 : 4 = 2
4
=
Proper fractions are fractions with the numerator less than
the denominator.
Improper fractions are fractions with the numerator greater
than or equal to the denominator.
1 ,
___
E.g.
4
2 , ___
9
___
... are proper
_____ fractions.
9
10
<
less than 1.
They are ______
Proper fractions are fractions with the numerator less than
the denominator.
Improper fractions are fractions with the numerator greater
than or equal to the denominator.
9 ,
___
E.g.
4
5 , ___
4 , ___
6
___
improper fractions.
... are ________
3
4
2
≥
greater than or equal to 1.
They are __________________
Which of the fractions above are equal to 1?
How can we recognize fractions which are equal to 1?
The numerator is equal to the denominator!!
Improper fractions are fractions with the numerator greater
than or equal to the denominator.
9 ,
___
E.g.
4
5 , ___
4 , ___
6
___
improper fractions.
... are ________
3
4
2
≥
greater than or equal to 1.
They are __________________
Which of the fractions above are greater than 1?
How can we recognize fractions which are greater than 1?
The numerator is greater than the denominator!!
Improper fractions are fractions with the numerator greater
than or equal to the denominator.
9 ,
___
E.g.
4
5 , ___
4 , ___
6
___
improper fractions.
... are ________
3
4
2
≥
greater than or equal to 1.
They are __________________
Any improper fraction can be changed into a mixed fraction or
a natural number.
Improper fractions are fractions with the numerator greater
than or equal to the denominator.
9 ,
___
E.g.
4
5 , ___
4 , ___
6
___
improper fractions.
... are ________
3
4
2
≥
Which of these fractions can be changed into mixed fractions ?
Change them (look at the picture)!
9
1
___
= 2 ___
4
4
2
5
___
___
= 1
3
3
Improper fractions are fractions with the numerator greater
than or equal to the denominator.
9 ,
___
E.g.
4
5 , ___
4 , ___
6
___
improper fractions.
... are ________
3
4
2
≥
Which of these fractions can be changed into natural numbers ?
Change them (look at the picture)!
4
___
= 1
4
6
___
= 3
2
Now, let's revise how to calculate it (without a picture)!
1.) Change into a mixed number or a natural number
a)
19
3
___
= 2 ___
8
8
Explanation:
19:8 equals 2 and remainder 3
Rewrite denominator!
=
(do what is possible):
Now, let's revise how to calculate it (without a picture)!
1.) Change into a mixed number or a natural number
a)
19
3
___
= 2 ___
8
8
b)
68
5
___
= 7 ___
9
9
c)
42
___
=
7
6
Explanation:
42:7 equals 6 (no remainder)
=
(do what is possible):
Now, let's revise how to calculate it (without a picture)!
1.) Change into a mixed number or a natural number
a)
b)
19
3
___
= 2 ___
8
8
68
5
___
= 7 ___
9
9
c)
42
___
=
7
6
d)
36
___
=
4
9
e)
2
___
=
9
(do what is possible):
How did we calculate
in all these tasks?
We divided the numerator
by the denominator.
Why?
Because the fraction line
always means division!
Now let's change a fraction
into a decimal number!
How to do it?
This is proper fraction
(numerator is less than
denominator),
We
should divide again,
so we can't change it into
mixedin writing...
buta now
fraction or into natural number!
Let's revise it...
Now, let's revise how to calculate it (without a picture)!
1.) Change into a mixed number or a natural number
a)
(do what is possible):
19
3
___
= 2 ___
8
8
19
___
= 19 : 8 = 2 .3 7 5
8
30
Remember:
60
So,
we
changed
same
fraction
0
Let's change the 4number
at
task
"a)"
intochange
athe
decimal
number...
When
we
fraction
into both
- mixed
fraction
How to do it?
into any
another
form,and
0
decimal
number.
we always
divide
(because the fraction line
means division)!
Only when we change into
decimal number,
then we use long division.
Now conversely! Let's revise how to change numbers
from other forms into fractions...
2.) Change into fraction:
6·8+7
a)
7
55
___
___
=
6
8
8
Rewrite
denominator!
=
Now conversely! Let's revise how to change numbers
from other forms into fractions...
2.) Change into fraction:
a)
7
55
___
___
=
6
8
8
b)
6
69
___
___
=
9
7
7
c)
8
16
24
___
___
___
8 =
=
=
= ...
1
2
3
(When we divide numerator by denominator, the result must be 8 !)
=
=
=
=
=
=
= ...
Now conversely! Let's revise how to change numbers
from other forms into fractions...
2.) Change into fraction:
a)
7
55
___
___
=
6
8
8
b)
6
69
___
___
=
9
7
7
c)
8
16
24
___
___
___
8 =
=
=
= ...
1
2
3
d)
241
2.41 = ____
100
2 decimal digits
2 zeros
=
Explanation:
Rewrite the given number, but without decimal point...
Write the digit 1 and as many zeros as we have
decimal digits in the given number...
Now conversely! Let's revise how to change numbers
from other forms into fractions...
2.) Change into fraction:
a)
7
55
___
___
=
6
8
8
f)
0.019 =
19
____
1000
b)
6
69
___
___
=
9
7
7
g)
27
54
27 = ___ = ___ = ...
1
2
c)
8
16
24
___
___
___
8 =
=
=
= ...
1
2
3
d)
241
2.41 = ____
100
h)
3
___
=
4
7
e)
309
30.9 = ____
10
i)
2893
28.93 = ____
100
31
___
7
Some decimal numbers can be changed into mixed fractions.
Let's revise it...
3.) Change into mixed number:
a)
41
2.41 = 2 ____
100
2 decimal digits
2 zeros
=
Recall:
2.41 can be changed not only into a
mixed fraction, but into an improper
fraction as well.
Say that improper fraction...
241
___
100
Some decimal numbers can be changed into mixed fractions.
Let's revise it...
3.) Change into mixed number:
a)
41
2.41 = 2 ____
100
b)
9
30.9 = 30 ____
10
c)
7
15.007 = 15 ____
1000
d)
0.045 =
This decimal number
can't be changed into a mixed fraction
because it has got zero wholes.
We can only change it into a fraction.
If we should change it into fraction,
45
the solution would be ____ .
1000
What does it mean - "to reduce a fraction" ?
To reduce a fraction means to divide both the numerator and
denominator of the fraction by the same number.
When we reduce a fraction, does the reduced fraction represent a
bigger or a smaller part than the original fraction?
They represent equal parts!!
When we reduce a fraction to the non-reducible fraction,
then we get this fraction in the lowest possible terms.
4.) Reduce these fractions to non-reducible fractions:
a)
5
10
5
___
= ___
12
6
6
=
2
We can reduce it by __.
So, we divide both the numerator and
denominator by 2 and write the results…
What does it mean - "to reduce a fraction" ?
To reduce a fraction means to divide both the numerator and
denominator of the fraction by the same number.
When we reduce a fraction, does the reduced fraction represent a
bigger or a smaller part than the original fraction?
They represent equal parts!!
When we reduce a fraction to the non-reducible fraction,
then we get this fraction in the lowest possible terms.
4.) Reduce these fractions to non-reducible fractions:
b)
4
24
4
___
= ___
30
5
5
=
6
We can reduce it by __.
What does it mean - "to reduce a fraction" ?
To reduce a fraction means to divide both the numerator and
denominator of the fraction by the same number.
When we reduce a fraction, does the reduced fraction represent a
bigger or a smaller part than the original fraction?
They represent equal parts!!
When we reduce a fraction to the non-reducible fraction,
then we get this fraction in the lowest possible terms.
4.) Reduce these fractions to non-reducible fractions:
c)
42
___
63
6
=
9
2
6
___
9
=
3
=
=
2
___
7 by __.
3
Now
We can
we can
reduce
reduce
it byagain,
__.
3
Now let's revise other properties of fractions...
5.) Complete these sentences:
four fourths.
a) One whole equals ____
4
___
4
1 =
=
twelve twelfths.
b) One whole equals _____
1 =
12
___
12
=
six thirds.
c) Two wholes equal ___
2 =
=
6
___
3
Now let's revise other properties of fractions...
5.) Complete:
4
d) 4 days = ___ week
7
Explanation:
7 days.
First, let's recall that a week has ___
1 of the week.
So we can conclude that each day is __
7
Now, we should think in this way:
1
__
of the week,
7
2
__
then 2 days are
of the week,
7
3
__
3 days are
of the week,
7
4
__
and 4 days are
of the week.
7
If 1 day is
Now let's revise other properties of fractions...
5.) Complete:
4
d) 4 days = ___ week
7
We can explain it in another way:
7 days.
Again, let's recall that a week has ___
So, let's consider it together with the rectangle divided into 7 equal parts!
week:
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
rectangle:
4? of the week
__
7
Left picture:
We are interested in
which part of the week consist of 4 days...
Right picture:
Which part of the rectangle
consist of 4 parts...
Now let's revise other properties of fractions...
5.) Complete:
4
d) 4 days = ___ week
7
Shortly:
Rewrite the given number into numerator.
In the denominator write the total number of days in a week !
As we already know,
the denominator is the total number of equal parts,
and the numerator describes the number of parts we are interested in.
Now let's revise other properties of fractions...
5.) Complete:
4
d) 4 days = ___ week
7
20
?
1
12
1
__
20
1
___
___
?
20 3
e) 20 min. =
hour =
hour
We
can reduce
9 it by3 __.
60
3
3
5
f) 5 months = ___ year
12
min.
hour
6
year:
January
February
March
April
May
June
July
August
September
Octobar
November
December
5
__
?
12
of the year
Now let's revise other properties of fractions...
6.) There are 7 tasks in the Ivan's homework.
Ivan solved 2 of them.
What portion of the homework did Ivan solve?
2
Ivan solved ___ of his homework.
7
What portion of the homework does he still have to solve?
He should solve
5
___
of his homework yet.
7
homework:
5
__
?
of the homework
7
1st task
2nd task
3rd task
4th task
5th task
6th task
7th task
2
__
? of the homework
7
Now let's revise other properties of fractions...
7
7.) If children ate ___ (seven tenths) of a cake, which
10
fraction of the cake remained?
3
___
(three tenths) of the cake remained.
10
3
___
of the cake
10
7
___
of the cake
10
Now let's revise other properties of fractions...
6
___
8.) Climber Dario climbed
of his path in one hour,
14
4
another ___
of the path in the next one hour,
14
3
and finally ___ of his path in the third hour.
14
Did he climb his whole path?
6
4
3
___
+ ___ + ___ =
14
14
14
13
___
14
No, he didn't climb his whole path.
There remained
1
___
of his path.
14
9.) Seven friends gathered some money and bought 3 chocolates
of equal size. They want to divide the chocolates equally.
How much chocolate will each of them get?
3
3 : 7 = ___
7
3
Each friend will get ___ of a chocolate.
7
Explanation with pictures:
When they divide the first chocolate into 7 equal parts,
1
each friend will get __ of the chocolate.
7
When they divide the second chocolate into 7 equal parts,
1
each friend will get __ of the chocolate.
7
When they divide the third chocolate into 7 equal parts,
1
each friend will get __ of the chocolate.
7
3
So, after all divisions each friend will have __ of a chocolate.
7
10.) a) 12 chocolates should be divided among 5 friends.
How many chocolates will each of them get?
2
12
12 : 5 = ___ = 2 ___
5
5
2
Each friend will get 2 ___ chocolates.
5
Explanation with pictures:
Each friend gets
1
__
of the 11th chocolate.
5
Each friend gets
1
__
of the 12th chocolate.
5
So, after all divisions each friend will have 2
2
__
chocolates!
5
10.) b) What about dividing 12 chocolates among 3 friends?
12 : 3 = 4
Each friend will get 4 chocolates.
Explanation with pictures:
After division each friend will have 4 chocolates.
12
11.) Little Ana ate ___ strawberries. How many strawberries
2
did she eat actually?
12
___
= 12 : 2 = 6
2
Little Ana ate 6 strawberries.
Explanation with pictures:
12
___
strawberries
2
= 6 strawberries
12.) There are 48 apples in the box.
3
5
___
of the box are red apples, ___ are green apples
8
12
and the rest of the apples are yellow.
a) How many apples are of which color?
red:
green:
yellow:
3
___
of 48 is
8
5
___
of 48 is
12
18 + 20 = 38,
18
( we calculated 48:8·3 )
20
48 - 38 = 10
There are 18 red, 20 green and 10 yellow apples in that box.
b) What portion of the box do yellow apples form?
5
5
___
10
5
Yellow
apples
form
(five twenty___
2 24
= ___
We can reduce it by __.
fourths) of the box.
48 24 24
13.) Complete these expressions:
a)
2
___
of 15 is 6
5
We calculated:
15 : 5 · 2 = 6
13.) Complete these expressions:
a)
2
___
of 15 is 6
5
How to explain this calculation?
2
Recall: If we want to color ___ of some figure, then
5
we divide it into 5 equal parts, and then color 2 parts.
Here we do just about the same thing!
2
___
of 15 can be calculated so that we divide
5
number 15 into 5 equal parts, and then take 2 parts.
15
: 5 · 2
=
6
13.) Complete these expressions:
a)
2
___
of 15 is 6
5
b)
4
___
of 72 is
9
32
c)
3
___
of 10 is
4
1
7 ___
2
E.g.
2
___
· 15
51
3
6
= ___ = 6
1
5
Both
procedures
We can
reduce it by __.
give equal results!!!
Here
we can't
so1 we must calculate in
3 word
15 10:4,
The
of5 divide
means
multiplication!
___
___
___
=
10
=
·
7
some another way!
42
2
2
2 in that way in the
can reduce
it by __.
Are weWe
allowed
to calculate
a and b tasks?
Yes, we are!!!
13.) Complete these expressions:
a)
2
___
of 15 is 6
5
b)
4
___
of 72 is
9
32
c)
3
___
of 10 is
4
1
7 ___
2
We got the same result !!!
The
denominator
tells
us
should
divide
this "bulk"
We
10 imagine
pieces
of
something,
10c?pears...
The
Howhave
numerator
can we
3 4tells
the
us
problem
tothat
takewe
in
3e.g.
of
part
these
4 parts...
into 4 equal parts...
1st
1
2
2nd
3
4
3rd
5
6
7
4th
8
How many pears do we have in these three parts?
9
10
7
1
__
2
13.) Complete these expressions:
a)
2
___
of 15 is 6
5
b)
4
___
of 72 is
9
32
c)
3
___
of 10 is
4
1
7 ___
2
d)
12
39
___
of ___ is
13
44
12
___
13
3
1
·
9
___
11
3
9
39
___ = ___
11
44
11
4
We
We can
can reduce
reduce it
it by
by __.
13
__.
14.) Complete:
a)
2
___
less
is ______
than 1 , by
5
3 .
___
5
Explanation with pictures:
2
___
of the rectangle is less than 1 rectangle,
5
<
3 of the rectangle.
by the uncolored part, and this part is ___
5
14.) Complete:
a)
2
___
less
is ______
than 1 , by
5
b)
19
___
greater than 1 , by
is _______
15
3 .
___
5
4 .
___
15
Explanation with pictures:
19
___
rectangles is greater than 1 rectangle,
15
>
by the part determined by second rectangle, and it is
4 of the rectangle.
___
15
14.) Complete:
a)
2
___
less
is ______
than 1 , by
5
b)
19
___
greater than 1 , by
is _______
15
3 .
___
5
4 .
___
15
15.) In each inequality below, which number is the greater?
a)
8
___
9
>
>
5
___
9
14.) Complete:
a)
2
___
less
is ______
than 1 , by
5
b)
19
___
greater than 1 , by
is _______
15
3 .
___
5
4 .
___
15
15.) In each inequality below, which number is the greater?
a)
b)
8
5
___
___
>
9
9
7
1
___
___
<
2
4
10
5
<
14.) Complete:
a)
2
___
less
is ______
than 1 , by
5
b)
19
___
greater than 1 , by
is _______
15
3 .
___
5
4 .
___
15
15.) In each inequality below, which number is the greater?
a)
b)
c)
8
5
___
___
>
9
9
7
1
___
___
<
2
4
10
5
3
___
4
11
<
<
5
14.) Complete:
a)
2
___
less
is ______
than 1 , by
5
b)
19
___
greater than 1 , by
is _______
15
3 .
___
5
4 .
___
15
15.) In each inequality below, which number is the greater?
a)
b)
c)
8
5
___
___
>
9
9
7
1
___
___
<
2
4
10
5
3
___
4
11
3
d) 6 ___
5
<
>
>
5
1
___
6
5
e)
8
5
___
·>· ___
3
2
16
>
15
We multiply
through diagonals...
Instead
of cross-multiplying,
we can find the common denominator
and then compare numerators...
If we would take the common
denominator 3·2, that is 6,
then we would get numerators
16 and 15. Multiplication through
diagonals is shortcut for that.
14.) Complete:
a)
2
___
less
is ______
than 1 , by
5
b)
19
___
greater than 1 , by
is _______
15
3 .
___
5
4 .
___
15
15.) In each inequality below, which number is the greater?
a)
b)
c)
8
5
___
___
>
9
9
7
1
___
___
<
2
4
10
5
3
___
4
11
3
d) 6 ___
5
<
5
>
1
___
6
5
e)
8
___
3
>
>
5
___
2
16.) Complete:
increases as well
a) If the numerator increases, then the fraction _____________.
E.g.
1
___
3
2
___
3
3
___
3
4
___
3
5
___
3
If we are looking from the left to the right,
increase
the numerators ________.
Colored parts, that is the fractions _____________.
increase as well
16.) Complete:
increases as well
a) If the numerator increases, then the fraction _____________.
decreases
b) If the denominator increases, then the fraction _________.
E.g.
1
___
1
1
___
2
1
___
3
1
___
4
1
___
5
If we are looking from the left to the right,
increase
the denominators ________.
Colored parts, that is, the fractions of the whole decrease
_______.
16.) Complete:
increases as well
a) If the numerator increases, then the fraction _____________.
decreases
b) If the denominator increases, then the fraction _________.
17.) Figure out:
a)
5
4
15 + 8
23
5
___
___
________
___
___
+
=
=
= 1
6
9
18
18
18
b)
4
1
1
9
1
27 - 1
26
___
- ___ = ___ - ___ = ________ = ___ =
2
6
2
6
6
6
2
___
= 4
6
1
3
1
= 4 ___
3
2
We can reduce it by __.
18.) Try to figure these out mentally!
a)
5
___
=
3 +
7
5
___
3
7
Explanation with pictures:
18.) Try to figure these out mentally!
a)
5
___
=
3 +
7
5
___
3
7
b)
3
1 - ___ =
5
2
___
5
Explanation with pictures:
18.) Try to figure these out mentally!
a)
5
___
=
3 +
7
5
___
3
7
b)
3
1 - ___ =
5
2
___
5
c)
2
7
___
___
=
5
6
9
9
Explanation with pictures:
18.) Try to figure these out mentally!
a)
5
___
=
3 +
7
5
___
3
7
b)
3
1 - ___ =
5
2
___
5
c)
2
7
___
___
=
5
6
d)
9
1
___
+ 5
4
2
9
1
= 9 ___
2
Explanation with pictures:
18.) Try to figure these out mentally!
a)
5
___
=
3 +
7
5
___
3
7
b)
3
1 - ___ =
5
2
___
5
c)
2
7
___
___
=
5
6
9
d)
1
___
+ 5
4
2
e)
6
2
___
- 6
11
9
1
= 9 ___
2
2
= ___
11
Explanation with pictures:
18.) Try to figure these out mentally!
a)
5
___
=
3 +
7
5
___
3
7
b)
3
1 - ___ =
5
2
___
5
c)
2
7
___
___
=
5
6
9
9
d)
1
___
+ 5
4
2
e)
6
f)
1
1
___
___
=
8
3
3
2
___
- 6
11
1
= 9 ___
2
2
= ___
11
8
Explanation with pictures:
18.) Try to figure these out mentally!
a)
5
___
=
3 +
7
5
___
3
7
b)
3
1 - ___ =
5
2
___
5
c)
2
7
___
___
=
5
6
9
9
d)
1
___
+ 5
4
2
e)
6
f)
1
1
___
___
=
8
3
3
g)
7
6 - 2 ___ =
8
2
___
- 6
11
1
= 9 ___
2
2
= ___
11
Explanation with pictures:
8
1
3 ___
8
19.) Figure out:
a)
b)
c)
3
3
27
21
9
2
___
___
___
___
·
=
= 1
49 7
9 1
7
7
9
We can reduce it by2__.
7
reduce
2We can___
54
___
·
=
4
9
19
6
38
54
12
___
___
___
=
= 12
·
9
19 1
1
1
19
We can reduce it by __.
9
can
1
9 3We ___
13 reduce
39it by __.
1
___
___
___
___
=
=
= 19
·
9 · 2
6
1
6 2
2
2
3
We can reduce it by __.
20.) Figure out:
a)
12
___
=
8 :
7
8 2 ___
7
14
2
___
___
___
= 4
·
=
1
12 3
3
3
4
We can reduce it by __.
b)
1
1
7
1
31
1
___
___
8
1
___
___
___
___
___
: 3
=
:
=
·
=
8
8
8
8
8
31
31
1
8
We can reduce it by __.
Let's recall the “number line”..
0
1
2
3
Place the following numbers onto the number line:
a)
5
___
2
6
2 and __
3
between __
4
Let's recall the “number line”..
0
1
2
3
Place the following numbers onto the number line:
a)
5
___
2
6
marked part of the number line
6 equal parts
should be divided into ____________
4
Let's recall the “number line”..
5
2 __
6
0
1
2
3
4
Place the following numbers onto the number line:
a)
5
___
2
6
5 parts of the whole from the left
we should count __
Let's recall the “number line”..
5
2 __
6
0
1
2
1
3 __
2
3
Place the following numbers onto the number line:
a)
5
___
2
6
b)
7
1
___
= 3 ___
2
2
middle
3 and __
4 , exactly in the ______
between __
7
__
2
4
Let's recall the “number line”..
5
2 __
6
0
1
2
1
3 __
2
3
7
__
2
4
Place the following numbers onto the number line:
a)
5
___
2
6
b)
7
1
___
= 3 ___
2
2
c)
2
___
3
it is not possible to change it into a mixed number,
there are no wholes,
0 and __
1
so this number lies between __
Let's recall the “number line”..
5
2 __
6
0
1
2
1
3 __
2
3
Place the following numbers onto the number line:
a)
5
___
2
6
b)
7
1
___
= 3 ___
2
2
c)
2
___
3
marked part of the number line
3 equal parts
should be divided into ____________
7
__
2
4
Let's recall the “number line”..
5
2 __
6
0
2
__
3
1
2
1
3 __
2
3
7
__
2
Place the following numbers onto the number line:
a)
5
___
2
6
b)
7
1
___
= 3 ___
2
2
c)
2
___
3
2 parts of the whole from the left
we count __
4
We shall continue this revision in writing...
Now we should be able to solve several more complex tasks
with more ‘fraction calculation’ operations.
Open your notebooks...
Author of presentation:
Antonija Horvatek
Croatia , October 2008.
With thanks to:
GSC
for support, great suggestions and
preliminary help with the translation
into English
and
Rex Boggs
for support and help with the
translation into fluent U.S. idiom
(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 Horvatek
http://public.carnet.hr/~ahorvate
[email protected]