Download Chapter 1-2, Supp. 1

1.1 Fractions: Defining Terms
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3
5
Fraction: part of a whole - example
Numerator: number on top
Denominator: number on bottom
Proper Fraction: numerator is less than the
denominator
Improper Fraction: numerator is equal to or
greater than the denominator
1.1 Problem Solving with Fractions
•
Mixed Number
 Consists of a whole number and a proper fraction
– example
1
2
3
1.2 Changing the Form of a Fraction
• Converting a mixed number to an improper
fraction: 3 3  3  8  3  27
8
8
8
• Converting an improper fraction to a mixed
number: 35
3
9
Divide 9 into 35:
35
8
3
9
9
9 35
27
8
1.2 Changing the Form of a Fraction
•
•
Multiplying or dividing the numerator
(top) and the denominator (bottom) of a
fraction by the same number does not
change the value of a fraction.
Writing a fraction in lowest terms:
1. Factor the top and bottom completely
2. Divide the top and bottom by the greatest
common factor
1.2 Changing the Form of a Fraction
• A number can be divided evenly by:
2 – if the last digit is 0, 2, 4, 6, 8
3 – if the sum of the digits is divisible by 3
4 – if the last two digits form a number that
is divisible by 4
5 – if the last digit is 0 or 5
6 – if the number is divisible by 2 and 3
1.2 Changing the Form of a Fraction
• A number can be divided evenly by:
7 – double the last digit and subtract it from
a number formed by the other digits. This
number must be zero or divisible by 7
8 – if the last three digits form a number
that is divisible by 8
9 – if the sum of the digits is divisible by 9
10 – if the last digit is 0
1.2 Changing the Form of a Fraction
• A prime number can only be divided evenly
by itself and the number 1
• Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19,
23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, etc.
• Factor trees can be used to factor a number
to its prime factorization
1.2 Changing the Form of a Fraction
– Factor Trees
18
6
3
2
3
1.3 Adding and Subtracting
Fractions
• Adding fractions with the same denominator:
a c
ac
 
b b
b
• Subtracting fractions with the same denominator:
a c
ac
 
b b
b
1.3 Adding and Subtracting
Fractions
•
To add or subtract fractions with different
denominators - get a common denominator.
• Using the least common denominator:
1. Factor both denominators completely
2. Multiply the largest number of repeats of each
prime factor together to get the LCD
3. Multiply the top and bottom of each fraction
by the number that produces the LCD in the
denominator
1.3 Adding and Subtracting Fractions –
no common factors in denominator
• Adding fractions with different denominators:
a c
ad  bc
 
b d
bd
• Subtracting fractions with different denominators:
a c
ad bc
 
b d
bd
1.3 Adding and Subtracting
Fractions
• Try these:
1 5
 ?
9 9
5
2

?
7 21
5 1
 ?
9 4
1.4 Multiplying and Dividing
Fractions
• Multiplying fractions:
a c
ac
 
b d
bd
• Dividing fractions:
a c
a d
ad
   
b d
b c
bc
1.4 Multiplying and Dividing
Fractions
• Complex Fractions: The numerator,
denominator or both are some sort of
fraction (proper, improper, or mixed)
• Example:
3
1
5  8  3  8  10  16  5 1
3
5 10 5 3
3
3
10
1.4 Multiplying and Dividing
Fractions
• Try these:
12
(simplify)
16
7 3

?
9 14
9
3
 ?
10 5
2.1 Reading, Writing, and Rounding
Decimals
• Place value: the position of the number in
relation to the decimal place
12345.67890
• What power of 10 does the 4 represent?
• What does the 8 represent?
• What about the 1?
2.1 Reading, Writing, and Rounding
Decimals
• Translating a decimal to words:
12.32
• In words: Twelve and thirty-two hundredths
• Translate the following:
37.281
2.1 Reading, Writing, and Rounding
Decimals
• Rounding a decimal:
– Look at the digit to the right of the place to
which you are rounding
– If the digit is less than 5 all the digits to the
right of the place you are rounding become
zero
– If the digit is 5 or greater, the place you are
rounding to is increased by 1 and all the digits
to the right of the place you are rounding
become zero
– Drop zeros to the right of the decimal place
2.1 Reading, Writing, and Rounding
Decimals
• Round 5.455 to the nearest tenth:
• 5 is next to the tenths place so increase 4 by
1 to get
5.500
• Drop the zeros:
5.5
2.2 Adding and Subtracting Decimals
• Write each number so that the decimal points
are in a vertical line
• Add the numbers as if there were no decimal
points.
• Place the decimal point in the answer in line
with the other decimal points
0.125
 1.25
 1.375
2.2 Adding and Subtracting Decimals
• Examples:
.0312
 10.1
 .6
 .56
 11.2912
10.125
 1.925
 8.2
2.3 Multiplying and Dividing Decimals
• To multiply decimals:
– Multiply the numbers as if there were no decimal points.
– Count the number of decimal places in each number and
add them together
– Put that many decimal places in the answer
0.125 1.2
Answer: 0.15
125
 12
250
1250
1500
2.3 Multiplying and Dividing Decimals
• To divide decimals:
– Write the numbers in long division format
– Move the decimal in the divisor to the right until you
have a whole number
– Move the decimal in the dividend to the right the
same number of places
– Divide as if the decimal points were not there
– Place the decimal in the answer just above the
decimal in the dividend
2.3 Multiplying and Dividing Decimals
• Example:
3.75
3
 8 30.00
0.8
24.
6.0
5.6
.40
.40
2.4 Converting Fractions and Decimals
• Converting decimals
to fractions: 0.125  125  25  5  1
1000
• Converting fractions
to decimals:
200
40
.375
3
 8 3.000
8
2.4
.60
.56
.040
.040
8
2.5 Converting Decimals and
Percents
• Write a decimal as a percent by moving the
decimal point 2 places to the right and
attaching a percent sign:
• Example:
0.382  38.2%
2.5 Converting Decimals and
Percents
• Write a percent as a decimal by moving the
decimal point 2 places to the left and
removing the percent sign:
• Example:
3.41  341%
2.6 Converting Fractions and
Percents
• Write a fraction as a percent by converting
the fraction to a decimal and then
converting the decimal to a percent:
.375
• Example:
3
8
 8
3.000
2.4
.60
.56
0.375  37.5%
.040
.040
2.6 Converting Fractions and
Percents
• Write a percent as a fraction by first
changing the percent to a decimal then
changing the decimal to the fraction and
reduce:
• Example:
45
95
9
45%  0.45 


100 20  5 20
Supplement: Chapter 1
1.1 Scientific Notation
•
Writing a number in scientific notation:
1. Move the decimal point to the right of the first nonzero digit.
2. Count the places you moved the decimal point.
3. The number of places that you counted in step 2 is the
exponent (without the sign)
4. If your original number (without the sign) was
smaller than 1, the exponent is negative. If it was
bigger than 1, the exponent is positive
Supplement: Chapter 1
1.1 Scientific Notation
• Converting to scientific notation (examples):
6200000  6.2 10?
.00012  1.2 10?
• Converting back – just undo the process:
6.203 1023  620,300,000,000,000,000,000,000
1.86 105  186,000
Supplement: Chapter 1
1.1 Scientific Notation
• Multiplication with scientific notation:
4 10  5 10   4  5 10
8
5
5
108 
 20 103  2 101 103  2 10 2
• Division with scientific notation:
4 10   4  10
5 10  5 10
12
12
4
4

.8 1012 4  .8 108  8 10 7
Supplement: Chapter 1
1.2 Uncertainty in Measurements
• Accuracy: correctness of a measurement
Example: The statue of liberty is 12.135
inches tall – the measurement is very
precise but inaccurate
• Precision: degree of correctness
Examples: 3.2 cm is more precise than 3 cm
but less precise than 3.24 cm
Supplement: Chapter 1
1.2 Uncertainty in Measurements
• Absolute error
Measurement
Absolute error
23 mg
0.5 mg
23.2 mg
0.05 mg
2.035 mg
0.0005 mg
Supplement: Chapter 1
1.2 Uncertainty in Measurements
• Lower limit = measurement – absolute error
• Upper limit = measurement + absolute error
• Relative error:
Absolute Error
Relative Error 
100%
Measurement
Supplement: Chapter 1
1.3 Estimation
• “” means “approximately equal to”
• Interval estimate: look at the first digit to get
the interval
Example: 347 + 231 + 583
Low Estimate: 300 + 200 + 500 = 1000
High Estimate: 400 + 300 + 600 = 1300
The actual sum is between 1000 and 1300
Supplement: Chapter 1
1.3 Estimation
• Rounding was covered in section 2.1 of the
text and can be used to find an estimate
• Example – find an estimate by rounding to the
tens place: 347 + 231 + 583
347 + 231 + 583  350 + 230 + 580 = 1160
An estimation is 1160