Name _____________________________________ Date ____________________ Period ______
Measurement Notes
Scientific Notation - an alternate way of writing numbers, usually with very large or very small
numbers. Generally scientific notation is NOT used when expressing
temperature, money, molecular masses and pH.
Consists of two parts:
Coefficient- A positive number greater than or equal to ____ and less than ____.
Exponent – A whole number exponent to which the base 10 is raised. Given the number 103,
the ____ is the exponent.
To convert a normal number to scientific notation:
Sample: Convert 159.0 to scientific notation
Sample: Convert -0.00634 to scientific notation
Practice 1: Convert the following numbers to scientific notation
___________________
1. 17600.0
___________________
5. 4.76
___________________
2. 0.00135
___________________
6. - 0.1544
___________________
3. 10.2
______________________7. 301.0
___________________
4. -67.30
___________________
8. -0.000130
To expand scientific notation to regular or normal notation:
1. If the exponent is positive, move the decimal to the ______ to make the number larger.
2. If the exponent is negative, move the decimal to the ______ to make the number smaller.
SAMPLE: Expand the scientific notation of 8.02 X 10-4 to regular notation
SAMPLE: Expand the scientific notation of -9.77 X 105 to regular notation
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Practice 2: Expand the following scientific notation to regular notation
_______________ 1. 4.96 X 10-2
_____________ 5. 7.01 X 100
________________ 2. 5.50 X 104
______________ 6. 5.61 X 10-3
_________________3. -9.3 X 10-3
______________7. 4.92 X 102
_________________4. -8.37 X 101
______________ 8. - 9.23 X 10-1
Every measurement we make is a comparison of the physical quantity being measured with a fixed
standard of measurement, such as the second or the centimeter. There are two words which
describe these comparisons--accuracy and precision. Understand the difference between them!
(1) ACCURACY - refers to how close a measurement is to the true or correct value for that
quantity.
(2) PRECISION refers to how close a set of measurements for a quantity are to
EACH OTHER.
- Is related to the size of the _______________ of the instrument. This tells you how
repeatable the measurements are or the amount of uncertainty in a measurement.
SAMPLE : A student weighs a beaker three times and records its mass as 45.21 g, 45.64g, and
45.73g.The actual mass of the beaker is 48.44 g.
Is this student accurate?
Is this student precise?
% error =
A student weighs a beaker and records its mass as 47.21. The actual mass of the beaker is
47.93g. What is the student’s % error?
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Significant figures (or digits) = are those digits in a measurement that can be determined _______
plus one that is estimated and is therefore uncertain.
You must be able to recognize, count, and round significant digits. There are basically two rules
about what a significant digit IS and IS NOT:
(1) all non-zero digits are ALWAYS significant, regardless of whether they occur
before or after a decimal point. For example: 446 has ____significant digits; 44.6 has
____ sig. figs and 0.446 has ____ sig figs.
(2) zeros have three different rules
(a) zeros at the beginning of numbers (leading zeros) are NEVER significant.
They are place holders only! For example, 0.678 has _____ sig figs; 0.0033 has
____ sig figs; 0.0004366 has ____ sig figs;
(b) zeros between non-zero digits (captive zeros) are ALWAYS significant. For
example, 706 has ____ sig figs; 711.004 has ____ sig figs; 30405.0066 has ____
sig figs
(c) zeros at the end of numbers (trailing zeros)are only significant if there is a
decimal point in the number or at the end of the number. For example, 760
has ____ significant figures, but 760. has ____ significant figures. 760.0 has
____ significant figures. The number 280,000 has ____ significant figures.
(3) Exact numbers: many times calculations involve numbers that were not obtained
using measured devices but were determined by counting. Example 10 experiments, 8
students, 6 molecules. These are exact numbers. All exact numbers have an infinite
number of significant figures.
Sample: Tell how many significant figures are in each of the following numbers:
_____ 20600
_____10 apples
_____ 0.00960
_____ 3.90 x 106
_____10.90
Practice 4: Tell how many significant digits are in each of the following numbers:
Pre-AP
____________1. 114.0
____________ 5. 4.50 X 103
____________2. 733.02
____________6. -23.40
____________ 3. 0.000310
____________7. 60 040
____________ 4. 9.080
____________8.
-0.0004010
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Significant Figures in Measurements
Three differently calibrated meter sticks can be used to measure the length of a board.
What is the measurement of each board?
Are there differences in the number of significant figures in the three measurements?
Which meter stick is the most accurate?
Rounding Significant Figures
Rounding to a set number of significant figures uses the standard convention for rounding, as
follows: a 5 or greater will round up to the next digit and a 4 or less will round down.
Example: Round 1234.5 to 4 sig figs
Round 0.06789 to 3 sig figs
Round 0.0034543003 to 3 sig figs
Round 1035600500 to 4 sig figs
Practice: Using standard rounding rules, round the following numbers to 3 significant digits.
Pre-AP
_______45.80
________ 135008
_______ 0.003295
________ 0.00536000
_______ 3.4239 x 105
________ 89.377
_______ 5.67 x 10-3
________2.2222 x 104
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Rules for math functions
1. Multiplication or division
*The answer should have the same number of ________ as the least precise
measurement.
Ex. 1.34 x 0.04 =
2. Addition or subtraction
*The answer should have the same number of _________________ as the least
precise measurement.
Ex. 1.34 + 2.3 =
Practice: Answer the following with the correct number of significant figures.
1) 6.2 x 10-4m + 5.70 x 10-3m =
2) 5.78 x 7.009 / 19.64 =
3) 1.82 x 10-5m * 9.40 x 106m * (7.8 +1.2009) =
4) 1.2578 x 0.0034 / (1.640 – 0.056) =
Density is the mass of a substance per unit volume or how much it weighs per given volume; it is
an intensive property (which means it is always the same for a substance unless it changes
phases) and therefore can be used for identification since it does not change with amount.
Putting the definition into a mathematical equation:
Density =
(1) mass is usually in grams;
(2) volume is usually in mL for liquids or cm3 for solids
(3) density is usually in units of g/mL or g/cm3
A MILLILITER and a CUBIC CENTIMETER ARE EXACTLY THE SAME AMOUNT of
volume!
These are just two different terms which mean the same volume and are sometimes used to
designate whether a substance is a solid or a liquid—ml for liquids and cm3 for solids.
However, particularly in medicine, liquids are almost always designated cm 3.

Pre-AP
The ________ object will be on the __________
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THE ONLY DENSITY WHICH MUST BE MEMORIZED IS WATER WHICH IS ________g/mL.
SUBSTANCES WHICH ARE LESS DENSE THAN WATER WILL ___________.
SUBSTANCES WHICH ARE MORE DENSE THAN WATER WILL ___________.
SAMPLE. A certain solid has a volume of 35.7 cm3 and weighs 85 grams. What is its density?
Practice: A liquid fills a graduated cylinder to the 45.6mL mark. The empty graduated cylinder
weighs 26.5g and when it is filled with an unknown liquid to the 45.6mL mark, the cylinder and the
liquid together weigh 70.0g. What is the density of the unknown liquid?
Practice:
If the density of gold is 19.3 g/cm3, what is the volume of 200g of gold in cm3?
Practice: You have 4 substances A-D. You know substance A is the most dense. Substance C is the
least dense and that substance D will float on substance B. Fill the graduated cylinder with the liquids
in the correct order based on their densities.
Practice:
grams?
The density of liquid mercury is 13.6g/mL. How much would 35.0 mL of mercury weigh in
Practice: Find the density of a rectangular solid whose dimensions are 3.4’’ by 1.2’’ by 1.7’’ if it is
known to weigh 1500g.
Practice: You have 4 substances Q-T. Using the information in the table fill the graduated cylinder
with the four substances and water.
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Practice: An irregular shaped stone was lowered into a graduated cylinder holding a volume of
water equal to 2.0mL. The height of the water rose to a volume of 7.0mL. If the mass of the stone was
25g, what was its density?
Buoyancy = _____________________________________________________________________
Viscosity = ______________________________________________________________________
Metric System
The metric unit of length is the meter, abbreviated m
The metric unit of volume is the liter, abbreviated L
The metric unit of weight (mass) is the gram, abbreviated g
The prefixes which you need to memorize NOW are the following
Prefix
Abbreviation
kilo
k
centi
c
milli
m
micro

nano
n
pico
p
Scientific Notation equivalent
Practice: What is the metric abbreviation for each of the following?
_____ (1) one-millionth of a gram
_____ (3) one thousand grams
_____ (2) one-thousandth of a meter
_____ (4) one-hundredth of a liter
We will learn to use a very organized and systematic method of problem solving known as
dimensional analysis.
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How to solve a problem using dimensional analysis.
(1) Draw the dimensional analysis grid like this.
(2) Put what is given in the UPPER LEFT corner of the grid (with label).
(3) Whatever label is in the upper left corner must be copied diagonally to the lower right corner.
This is arranging your labels correctly for canceling later.
(4) Put the label of what you are converting TO in the UPPER RIGHT corner.
(5) Fill in a correct numerical relationship which exists between the 2 labels.
(6) Cancel any labels which appear both in the numerator and the denominator of the grid.
(7) Multiply everything together that is above the grid line; divide by everything that is below the grid
line.
(8) Express your answer in the same number of significant figures as were given in the ORIGINAL
problem.
(9) BE SURE YOUR ANSWER HAS A LABEL.
Hint: When using dimensional analysis to convert the metric system, a good rule
of thumb to follow until you become more comfortable with using dimensional analysis
and the metric system is to always put the "1" in front of the prefix (in whichever
dimension the prefix is—top or bottom) and then simply define that prefix in scientific
notation in the other part of the dimension. Don't make rules about always putting the "1"
on the top of the dimension or on the bottom because this will not work.
SAMPLE: Convert 3.179 hours to minutes
SAMPLE: Convert 1.43 meters to kilometers
Practice:
Practice:
Pre-AP
Convert 378.4 cm to meters
Convert 4.32 X 10-4 g to micrograms
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WHAT IF YOU NEED TO CONVERT FROM PREFIX TO PREFIX--HOW DO YOU KNOW WHERE
TO PUT THE "1"? (first, use one dimension to convert from the given prefix to just the plain unit, or
base, draw another dimension and then convert from the unit to the other prefix). Be sure to arrange
things so that every label will cancel out except what you are looking for.
SAMPLE : Convert 231 mm to nm.
Practice: Convert 54.7 micrograms to kilograms
Practice: Convert 5.43 cL to mL.
Practice:
How many nanometers are there in 45.2 kilometers?
You will use dimensional analysis to convert the metric system to the American system and vice
versa.
These are the American conversions I expect you to know all others would be given to you.
1 ton
=
1 pound
_______ pounds
1 quart
= _______ ounces
= _______ pints
1 gallon = _______ quarts
1 mile
= _______ feet
1 day
1 foot
= _______ inches
1 hour
1 yard
= _______ feet or _______ inches
1 minute = _______seconds
1 pint = _______ cups
1 year
= _______ hours
= _______ minutes
= _______ days
METRIC TO AMERICAN CONVERSIONS
2.54 cm
=
1.06 quarts
454 grams
Pre-AP
1 inch
=
=
(length)
1 liter
(volume)
1 pound
(mass)
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SAMPLE: Convert 4.10 miles to millimeters
Practice 14:
Convert 3.92 yards to picometers.
Practice 15: Convert 1.75 X 106 grams to ounces
Practice 16: Convert 145 lb to pg
Practice 17: Convert 1.96 X 10-3 kilometers/hour to inches/second
Practice 18: Convert 10.6 gal/min to L/sec
Practice 19: Convert 0.008888 tons/0.500 day to mg/sec
Practice 20: If a pitcher throws a baseball at 95.5 miles per hour, what is this in inches per minute?
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