Welcome to Seminar 2Agenda
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 .
The denominator tells us how many congruent pieces the whole is divided into.
The numerator tells us how many such pieces are being considered.
numerator
denominatorba
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
2
1=
4
2
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,
2
1=
4
2=
6
3
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.
2
1=
4
2=
6
3=
8
4
Therefore,
The Whole1/2 1/21/3 1/3 1/31/4 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 1/71/8 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 both 6 and 10. How does this look?6/2 = 3 10/2= 5 6/10=3/5
is not reduced because ??? divides into both 35 and 40.
10
6
40
35
How do we know that two fractions are the same?
.
260
110
15
8
23
11
To find out whether two fraction are equal, we need to reduce them to their lowest terms or simply…
40
35
35/5= 7 40/5= 8 35/40=7/8
260
110
How do we know that two fractions are the same?
Are21
14 and 45
30 equal?
21
14 reduce3
2
721
714
45
30 reduce9
6
545
530 reduce
3
2
39
36
Now we know that these two fractions are actually the same!
21
14
45
30=
Improper Fractions and Mixed Numbers
An improper fraction can be converted to a mixed number and vice versa.
3
5An improper fraction is a fraction with the numerator larger than or equal to the denominator.
A mixed number is a whole number and a fraction together 7
32
Any whole number can be transformed into an improper fraction.
,1
44
Improper Fractions and Mixed Numbers
3
21
3
5
Converting improper fractions into mixed numbers:- divide the numerator by the denominator - the quotient is the leading number,- the remainder as the new numerator.-Denominator stays the same
4
7
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.
,4
31
4
7More examples:
Converting mixed numbers into improper fractions.
7
17
7
17314
1472
7
32
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
5
43
Converting mixed numbers into improper fractions.
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
5
19
5
453
5
43
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 different denominatorsIn this case, we need to first convert them into equivalent fraction with the same denominator.Example:
5
2
3
1
An easy choice for a common denominator is 3×5 = 15Now we have our same denominator
1515
15
?
3
1 + 15
?
5
2 515
53/15
x
15
5
623
35/15
x
15
6+ 15
11
Addition of Fractions with different denominatorsIn this case, we need to first convert them into equivalent fraction with the same denominator.Example:
15
5
53
51
3
1
15
6
35
32
5
2
5
2
3
1
An easy choice for a common denominator is 3×5 = 15Now we have our same denominator
Therefore,
15
11
15
6
15
5
5
2
3
1
1515
15
?
3
1 +
15
?
5
2 615
53/15
15
6
523
35/15
15
5
More Exercises:
7
2
5
3
9
4
6
5
=
=
57
52
75
73
=35
10
35
21
35
31
35
1021
=
7
2
5
3
35
?
35
?
5x72 primes
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
More examples:
23
11
10
7
1023
1011
2310
237
2310
1011237
230
110161
230
51
Examples of dividing fractionsExamples of dividing fractions
5 10
9 12
KEEPSWITCH to
multiply
FLIP number following the division
sign
59
x 12
10
10
12
9
5x
9
6
90
603
2
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?
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?
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 ten-thousandths.”
• 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
Slide 3- 49Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
To convert from decimal to fraction notation,• a) count the number of decimal
places,
• b) move the decimal point thatmany places to the right, and
• c) write the answer over a denominator with a 1 followed by that number of zeros
4.98
4.98
2 zeros
2 places
Move
2 places.
498
100
Slide 3- 50Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
• Write fraction notation for 0.924. Do not simplify.
• 0.924 =
Slide 3- 51Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
• Write fraction notation for 0.924. Do not simplify.
• Solution
9240.
0924
100
3 places
3 zeros
0.924.
Slide 3- 52Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
•Write 17.77 as a fraction and as a mixed numeral.
Slide 3- 53Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
•Write 17.77 as a fraction and as a mixed numeral.
•Solution•To write as a fraction:
•17.77 177717
0.77
10
2 zeros
2 places
17.77
7717.77 17
100
To write as a mixed numeral, we rewrite the whole number part and express the rest in
fraction form:
Slide 3- 54Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
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
b) move the decimal point that
number of places to the left. Leave
off the denominator.
8.679.Move
3 places.
3 zeros
8679
1000
86798.679
1000
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 5 6
Slide 3- 56Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
Example D
• Subtract 574 – 570.175
• Solution
5 7 4 . 0 0 0 57
3 . 8 2 5
7 0 .5
9 9 103
1
Slide 3- 57Copyright © 2008 Pearson
Education, Inc. Publishing as Pearson Addison-Wesley
To multiply using decimals: 0.8 0.43
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).
0.43
3
0.8
4 4
2
0.43
0.8
0.344
(2 decimal places)
(1 decimal place)
(3 decimal places)
Ignore the decimal points for now.