welcome to seminar 2 agenda questions about last week discussion/mml reminder fraction basics week 2...

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


Are

21

14 and 45

30 equal?


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


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


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.


5

43


Think about the order of operations. PEMDAS what comes first? Multiply the denominator by the whole number, then add the numerator.


5

19

5

453

5

43


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:83

81


More examples

5

1

5

2


More examples

5

1

5

2

5

3

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: What is the lowest common multiple?

8

1

4

3 =

8

1

24

23

8

1

4

3 = =

8

1

8

6 =

8

7

8

16

8

1

4

3

8

1

8

?= 8/4=2 2x3=6

8

6

More Exercises: 2 primes

7

2

5

3 =

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

More Exercises:

9

4

6

5 =

69

64

96

95

=54

24

54

45

54

151

54

69

54

2445

=

Adding Mixed Numbers

Example:5

32

5

13

5

32

5

13

5

3

5

123

5

315

5

45

5

45


Another Example:

8

31

7

42


Another Example:

8

31

7

42

8

31

7

42

8

3

7

412

56

73843

56

533

56

533

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:12

3

12

11


More examples:

9

4

7

6


More examples:

9

4

7

6

79

74

97

96

63

28

63

54

63

2854 26

63

23

11

10

7


More examples:

23

11

10

7

1023

1011

2310

237

2310

1011237

230

110161

230

51

Examples of multiplying fractionsExamples of multiplying fractions

2 ½ X ¼

5 1

2 4X

5

8

9

4

7

6x

9

4

7

6x

11

8

63

24

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

9

4/

7

6

4

9

7

6x

28

241

28

547

61

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 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

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



• Write fraction notation for 0.924. Do not simplify.

• 0.924 =



• Write fraction notation for 0.924. Do not simplify.

• Solution

9240.

0924

100

3 places

3 zeros

0.924.



•Write 17.77 as a fraction and as a mixed numeral.



•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:



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



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



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.

Unit 2

• Participate in Discussion

• Complete Quiz

• Practice in MML

• Complete Reading

• Practice Problems

welcome to seminar 2 agenda questions about last week discussion/mml reminder fraction basics week 2...

Documents