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Welcome to Seminar 2
Agenda
Questions about last week
Discussion/MML Reminder
Fraction Basics
Week 2 Overview
Questions
Definition:
A fraction is an ordered pair of whole numbers, the 1st one
is usually written on top of the other, such as ½ or ¾ .
a
b
numerator
denominator
The denominator tells us how many congruent pieces
the whole is divided into.
The numerator tells us how many such pieces are
being considered.
Same
measure/
size
Equivalent fractions
a fraction can have many different appearances, these
are called equivalent fractions
In the following picture we have ½ of a cake
because the whole cake is divided into two congruent
parts and we have only one of those parts.
But if we cut the cake into smaller
congruent pieces, we can see that
1
2
=
2
4
Or we can cut the original cake
into 6 congruent pieces,
Equivalent fractions
a fraction can have many different appearances, these
are called equivalent fractions
Now we have 3 pieces out of 6 equal pieces, but
the total amount we have is still the same.
Therefore,
1
2
3
=
=
2
4
6
If you don’t like this, we can cut
the original cake into 8 congruent
pieces,
Equivalent fractions
a fraction can have many different appearances, they
are called equivalent fractions
then we have 4 pieces out of 8 equal pieces, but the
total amount we have is still the same.
Therefore,
1
2
3
4
=
=
=
2
4
6
8
The Whole
1/2
1/2
1/3
1/3
1/3
1/4
1/4
1/4
1/4
1/5
1/5
1/5
1/5
1/5
1/6
1/6
1/6
1/6
1/6
1/6
1/7 1/7 1/7 1/7 1/7 1/7 1/7
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
How do we know that two fractions are the same?
we cannot tell whether two fractions are the same until
we reduce them to their lowest terms.
A fraction is in its lowest terms (or is reduced) if we
cannot find a whole number (other than 1) that can divide
into both its numerator and denominator.
Examples:
is not reduced because 2 can divide into
6
both 6 and 10. How does this look?
10
6/2 = 3 10/2= 5
6/10=3/5
35
40
is not reduced because ??? divides into
both 35 and 40.
How do we know that two fractions are the same?
35
40
35/5= 7
40/5= 8
35/40=7/8
110
260
8
15
110
260
.
11
23
To find out whether two fraction are equal, we need to
reduce them to their lowest terms or simply…
How do we know that two fractions are the same?
Are
14
21
and
30
45
equal?
How do we know that two fractions are the same?
Are 14
21
and
14
21
reduce
30
45
reduce
30
45
equal?
14  7 2

21  7 3
30  5 6

45  5 9
14
30
=
21
45
reduce
63 2

93 3
Now we know that these two fractions are actually
the same!
Improper Fractions and Mixed Numbers
An improper fraction is a fraction
with the numerator larger than or
equal to the denominator.
5
3
Any whole number can be
transformed into an improper
fraction.
4
4 ,
1
A mixed number is a whole
number and a fraction together
3
2
7
An improper fraction can be converted to a mixed
number and vice versa.
Improper Fractions and Mixed Numbers
Converting improper fractions into
mixed numbers:
5
2
1
3
3
- divide the numerator by the denominator
- the quotient is the leading number,
- the remainder as the new numerator.
-Denominator stays the same
7

4
Improper Fractions and Mixed Numbers
Converting improper fractions into
mixed numbers:
- divide the numerator by the denominator
- the quotient is the leading number,
- the remainder as the new numerator.
More examples:
7
3
1 ,
4
4
Converting mixed numbers into improper fractions.
2  7  14
3 14  3  17 17
2 

7
7
7
Think about the order of
operations. PEMDAS what
comes first? Multiply the
denominator by the whole
number, then add the
numerator.
Improper Fractions and Mixed Numbers
Think about the order of operations.
PEMDAS what comes first?
Multiply the denominator by the
whole number, then add the
numerator.
Converting mixed numbers
into improper fractions.
4
3
5
Improper Fractions and Mixed Numbers
4 3  5  4 19
3 

5
5
5
Converting mixed numbers
into improper fractions.
Addition of Fractions
- addition means combining objects in two or
more sets
- the objects must be of the same type, i.e. we
combine bundles with bundles and sticks with
sticks.
- in fractions, we can only combine pieces of the
same size. In other words, the denominators
must be the same.
Addition of Fractions with equal denominators
Example: 1  3
8 8
+
= ?
Addition of Fractions with equal denominators
More examples
2 1
 
5 5
Addition of Fractions with equal denominators
More examples
2 1
3
 
5
5 5
Addition of Fractions with
different denominators
In this case, we need to first convert them into equivalent
fraction with the same denominator.
Example:
1 2

3 5
An easy choice for a common denominator is 3×5 = 15
Now we have our same denominator

15 15
1 ?

3 15
2 ?
+ 5  15
15 / 5  3
3x2  6
15 / 3  5
5 x1  5
5
15
+
6 11

15 15
Addition of Fractions with
different denominators
In this case, we need to first convert them into equivalent
fraction with the same denominator.
Example:
1 2

3 5
An easy choice for a common denominator is 3×5 = 15
Now we have our same denominator

1 ?

3 15
+
2 ?

5 15
6
15
2 23 6


5 5  3 15
1 2 5 6 11
   
3 5 15 15 15
1
1 5
5


3
3 5
15
Therefore,
15 / 3  5
5 1  6
15 15
15 / 5  3
3 2  5
5
15
More Exercises: What is
the lowest common
multiple?
3 1

=
4 8
3 1

4 8
=
? 1

8 8
3 1
3 2 1
6 1
6 1 7
 =
 = 

=
4 8
4 2 8
8 8
8
8
8/4=2
2x3=6
6
8
More Exercises: 2 primes
3 2
 =
5 7
More Exercises:
3 2

5 7
?
?

35 35
5x7
2 primes
3 2
3 7 2  5
21 10
 =


=
5 7
5 7 7  5
35 35
5 4
 =
6 9
21  10 31

=
35
35
More Exercises:
5 4
5 9 4 6
45 24
 =


=
6 9
69 9 6
54 54
45  24 69
15


1
=
54
54
54
Adding Mixed Numbers
Example:
1
3
1
3
3  2  3  2
5
5
5
5
1 3
 3 2 
5 5
1 3
 5
5
4
 5
5
4
5
5
Adding Mixed Numbers
Another Example:
4 3
2 1
7 8
Adding Mixed Numbers
Another Example:
4 3
4
3
2 1  2  1
7 8
7
8
4 3
 2 1 
7 8
4  8  3 7
 3
56
53
53
 3
3
56
56
Subtraction of Fractions
- subtraction means taking objects away.
- the objects must be of the same type, i.e. we
can only take away apples from a group of
apples.
- in fractions, we can only take away pieces of
the same size. In other words, the denominators
must be the same.
Subtraction of Fractions with equal denominators
Example: 11  3
12 12
Subtraction of Fractions
More examples:
6 4
 
7 9
Subtraction of Fractions
More examples:
6 4
69 4 7
54 28 54  28
26
 





7 9
79 9 7
63 63
63
63
7 11


10 23
Subtraction of Fractions
More examples:
7 11
7  23 1110
7  23  1110
161 110





10 23 10  23 23 10
10  23
230
51

230
Examples of multiplying fractions
2½
X
¼
5
2
X
1
5
4
8
6 4
x 
7 9
6 4 24 8
x 

7 9
63 11
Examples of dividing fractions
5
9
KEEP
5
10
9
12
12
x
SWITCH to
multiply
5 12 60 6
x   
9 10 90 9
FLIP number
following the division
sign
2
3
10
6 4
/ 
7 9
6 9 54
6
24
x 
1
1 
7 4 28
7
28
For Project Unit 3
• A recipe for a drink calls for 1/5 quart water
and ¾ quart apple juice.
• How much liquid is needed?
• 2/5 + 1/4 = 8/20 + 5/20 = 13/20
• Now if the recipe is doubled?
13/20
• 13/20 + 13/20 = 26/20 =1 6/20= 1 3/10
• Or
• 13/20 * 2 = 13/20 *2/1 =26/20 = 1 6/20 =
1 3/10
If the recipe is halved?
13/20
• 13/20 / 2 = 13/20 / 2/1 = 13/20 * ½= 13/40
42.3245
• 4 tens + 2 ones + 3 tenths + 2 hundredths + 4 thousandths + 5 ten-
thousandths
• We read this number as
• “Forty-two and three thousand two hundred forty-five tenthousandths.”
• The decimal point is read as “and”.
•Write a word name for the number in this
sentence: The top women’s time for the 50
yard freestyle is 22.62 seconds.
•Write a word name for the number in this
sentence: The top women’s time for the 50 yard
freestyle is 22.62 seconds.
•Twenty-two and sixty-two hundredths
To convert from decimal to fraction notation,
• a) count the number of decimal
places,
4.98
2 places
• b) move the decimal point that
many places to the right, and
4.98
Move
2 places.
• c) write the answer over a
denominator with a 1 followed
by that number of zeros
Slide 3- 49
Copyright © 2008 Pearson
Education, Inc. Publishing as
498
100
2 zeros
• Write fraction notation for 0.924. Do not
simplify.
• 0.924 =
Slide 3- 50
Copyright © 2008 Pearson
Education, Inc. Publishing as
• Write fraction notation for 0.924. Do not
simplify.
• Solution
0.924.
3 places
924
0.924 
1000
3 zeros
Slide 3- 51
Copyright © 2008 Pearson
Education, Inc. Publishing as
•Write 17.77 as a fraction and as a mixed
numeral.
Slide 3- 52
Copyright © 2008 Pearson
Education, Inc. Publishing as
•Write 17.77 as a fraction and as a mixed numeral.
•Solution
•To write as a fraction:
•17.77
17.77
2 places
1777
17.77 
100
2 zeros
To write as a mixed
numeral, we rewrite the
whole number part and
express the rest in
77
17.77  17
100
fraction form:
Slide 3- 53
Copyright © 2008 Pearson
Education, Inc. Publishing as
To convert from fraction notation to decimal notation
when the denominator is 10, 100, 1000 and so on,
a) count the number of zeros, and
8679
1000
3 zeros
b) move the decimal point that 8.679.
Move
number of places to the left. Leave 3 places.
off the denominator.
8679
 8.679
1000
Slide 3- 54
Copyright © 2008 Pearson
Education, Inc. Publishing as
Add: 4.31 + 0.146 + 14.2
Solution Line up the decimal
points and write extra zeros
4. 3 1 0
. 1 4 6
1 4 . 2 0 0
1 8. 6 56
Example D
• Subtract 574 – 570.175
• Solution
3
9
9
10
5 7 4 . 0 0 0
5 7 0 . 1 7 5
3 . 82 5
Slide 3- 56
Copyright © 2008 Pearson
Education, Inc. Publishing as
To multiply using decimals:
a) Ignore the decimal points,
and multiply as though both
factors were whole numbers.
b) Then place the decimal point in
the result. The number of decimal
places in the product is the sum of the
number of places in the factors.
(count places from the right).
Slide 3- 57
Copyright © 2008 Pearson
Education, Inc. Publishing as
0.8  0.43
2
0.43
 0.8
344
Ignore the
decimal points
for now.
0.43
 0.8
(2 decimal places)
0.344
(3 decimal places)
(1 decimal place)
Unit 2
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•
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