Teacher Resources Needed for Lesson
• Copies of the following worksheets:
– Rounding with Addition and Subtraction
– Rounding with Multiplication and Division
– Scientific Notation
– Significant Digit Worksheet
Rounding Answers and Scientific
Notation
Calculating with Significant Digits
• In this course you will often take
measurements and use them to calculate
other quantities
• Must be careful to keep track of which digits in
your calculations and results are significant
• We do this because your results of your
calculations should not imply more certainty
than your measured quantities justify
Example
• You are required to calculate the volume of a box.
The dimensions of the box are determined to be
12.2cm by 10.1 cm by 9.40 cm.
• When you do your calculation you get:
𝑉 = 𝑙𝑤ℎ
𝑉 = 12.2𝑐𝑚 10.1𝑐𝑚 9.40𝑐𝑚
𝑉 = 1158.268𝑐𝑚3
• The measurement all have three significant digits
but the answer is reported to seven significant
digits. This is saying that you are more certain of
the value of the volume than you are of the initial
measurements. This is not possible.
Rules for Reporting Significant Digits in
Calculations
• Rule 1: Multiplying and Dividing
The value with the fewest number of significant
digits, going into the calculation, determines the
number of significant digits that you should report
in your answer.
• Rule 2: Adding and Subtracting:
The value with the fewest number of decimal
places, going into the calculation, determines the
number of decimal places that you should report in
your answer.
Rules for Reporting Significant Digits in Calculations
• Rule 3: Rounding
To get the appropriate number of significant digits (rule 1)
or decimal places (rule 2), you may need to round your
answer.
 If the first digit to be dropped is 4 or less, the preceding
digit is not changed. (example: 8.674 is rounded to
8.67)
 If the first digit to be dropped is 5, the preceding digit
is increased by 1. (example: 8.675 123 is rounded up to
8.68)
 If the first digit to be dropped is a lone 5 or a 5
followed by zeros, the preceding digit is not changed if
it is even, but is increased if it is odd. (example: 8.675 is
rounded up to 8.68, but 8.665 is rounded down to
8.66)
 When solving multistep problems record all the digits
until the final answer to avoid rounding error
Practice Problem (as a class)
• A student measured a rectangular shaped
sample of iron and found it to be 6.78 cm
long, 3.906 cm wide, and 11 cm tall.
Determine its volume to the correct number
of significant digits.
Practice Problem (as a class)
• Suppose that you measure the masses of four
objects as 12.5 g, 145.67 g, 79.0 g, and 38.438
g. What is the total mass of the objects?
Homework!!!
• Complete the following worksheets:
– Rounding with addition and subtraction
– Rounding with multiplication and division
Scientific Notation
Scientific Notation
• Scientific notation expresses a number by
writing it in the form 𝑎 × 10𝑛 , where 1 ≤
𝑎 < 10
• Example: 2.997 924 58 × 108 𝑚/𝑠
• Digits in the coefficient 𝑎 are all significant
• This makes it easier to write extremely large
and extremely small numbers
Converting Numbers into Scientific
Notation
• To convert a number into scientific notation you must first move the
decimal place until the number is 1 ≤ 𝑎 < 10
• Count the number of times you move the decimal place and
whether it has been moved to the right or the left
• If the decimal is moved to left 𝑥 times then the power is 𝑎 × 10𝑥
Example 1:
Convert 20 580 into scientific notation.
1)
2)
3)
Move decimal place to the left four times
2.0580
Place the number of times you moved the decimal place as your
power of ten
2.0580 × 104
If you are unsure if you did it correctly move the decimal place the
number of times indicated in the power and if you end up with
the number you started with originally you have done it correctly
Converting Numbers into Scientific
Notation
• To convert a number into scientific notation you must first move the
decimal place until the number is 1 ≤ 𝑎 < 10
• Count the number of times you move the decimal place and
whether it has been moved to the right or the left
• If the decimal is moved to right 𝑥 times then the power is 𝑎 × 10𝑥
Example 2:
Convert 0.00000546 into scientific notation.
1)
Move decimal place to the right six times
5.46
2) Place the number of times you moved the decimal place as your
power of ten
5.46 × 10−6
(notice the negative sign when the decimal is moved to the right)
Practice Problems
• Copy the following into your notes and convert
them into scientific notation:
a)
b)
c)
d)
e)
f)
9 523 458
0.000 001 952
756 300 000 000
0.000 000 000 000 123
15
0.23
Practice Problems
• Copy the following examples into your notes
and convert them into standard notation:
a)
b)
c)
d)
e)
1.235 × 107
5.68 × 10−5
7.896 54 × 100
4.52 × 1012
8.889 × 10−8
Calculations Involving Scientific
Notation
• Can enter these values into a calculator and calculate an
answer
• If you are not using a calculator there are some rules you
can follow to make calculations easier
Addition and Subtraction:
• Change all the numbers to the same power of 10 and then
add or subtract the numbers
Example:
1.234 × 105 + 4.2 × 104 = 1.234 × 105 + 0.42 × 105
= 1.234 + 0.42 × 105
= 1.654 × 105
= 1.65 × 105
Calculations Involving Scientific
Notation
Multiplication and Division:
• Multiply or divide the coefficients, add or subtract the
exponents, and express the result in scientific notation
General Form for Exponents:
𝑥 𝑎 𝑥 𝑏 = 𝑥 𝑎+𝑏
𝑥𝑎
𝑎−𝑏
=
𝑥
𝑥𝑏
Example 1:
1.36 × 104 3.76 × 103 = 5.11 × 107
Example 2:
4.51 × 105
9
=
0.572
×
10
7.89 × 10−4
= 5.72 × 108
Homework!!!
• Complete the following worksheets:
– Scientific Notation
– Significant Digit Worksheet