Measurement. How far, how much, how many?. PROBLEM SOLVING. STEP 1: Understand the Problem STEP 2: Devise a Plan STEP 3: Carry Out the Plan STEP 4: Look Back. Step 1. Understand the Problem. Step 2. Devise a Plan. Step 3. Carry Out the Plan. Step 4. Look Back. A Number A Quantity
How far, how much, how many?
STEP 1: Understand the Problem
STEP 2: Devise a Plan
STEP 3: Carry Out the Plan
STEP 4: Look Back
An implied or measured quantity has significant figures associated with the measurement
1 mile = 1603 meters
Exact - defined measured - 4 sig figs
An exact number is not measured, it is defined or counted; therefore, it does not have significant figures or it has an unlimited number of significant figures.
1 kg = 1000 grams
1.0000000 kg = 1000.0000000 grams
Quantitative- use numbers to describe measurement– test equipment, counts, etc.
Qualitative- use descriptions without numbers to descript measurement- use five senses to describe
Quantitative- easy check
Easy to agree upon, no personal bias
The measuring instrument limits how good the measurement is
All measurements contain some uncertainty.
Uncertainty is measured with
Accuracy How close to the true value
Precision How close to each other
Measures how close the experimental measurement is to the accepted, true or book value for that measurement
Is the description of how good that measurement is, how many significant figures it has and how repeatable the measurement is.
Accuracy can be true of an individual measurement or the average of several
Precision requires several measurements before anything can be said about it
We can’t say!
Synonyms for Accuracy…
The number of significant digits is independent of the decimal point.
All have three
Imply how the quantity is measured and to what precision.
Are always dependant upon the equipment or scale used when making the measurement
0.2, 0.3, 0.4?
0.2, 0.3, 0.4?
0.26, 0.27, or 0.28?
0.2, 0.3, 0.4?
0.26, 0.27, or 0.28?
0.262, 0.263, 0.264?
Leading zeros are notsignificant.
0.00421 - three significant figures
4.21 x 10-3
Notice zeros are not written in scientific notation
Captive zeros are significant.
4012 - four significant figures
4.012 x 103
Notice zero is written in scientific notation
Trailing zeros before the decimal are notsignificant.
4210000 - three significant figures
Trailing zeros after the decimal are significant.
114.20 - five significant figures
Zeros are what will give you a headache!
They are used/misused all of the time.
The press might report that the federal deficit is three trillion dollars. What did they mean?
$3 x 1012
Scientific notation - can be used to clearly express significant figures.
A properly written number in scientific notation always has the the proper number of significant figures.
0.003210 = 3.210 x 10-3
The accuracy is measured by comparing the result of your experiment with a true or book value.
The block of wood is known to weigh exactly 1.5982 grams.
The average value you calculated is 1.48 g.
Is this an accurate measurement?
Is used to write very, very small numbers or very large numbers
Is used to imply a specific number of significant figures
Uses exponentials or powers of 10
large positive exponentials imply numbers much greater than 1
negative exponentials imply numbers smaller than 1
Format is Mantissa x Base Power
Decimal part of
We just move the decimal point around.
If a number is larger than 1
1 2 3 0 0 0 0 0 0 = 1.23 x 108
If a number is smaller than 1
0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10-7
Most scientific calculators use scientific notation when the numbers get very large or small.
How scientific notation is
displayed can vary.
It may use x10n
or may be displayed
using an E or e.
They usually have an Exp or EE
button. This is to enter in the exponent.
3.78 x 10 5
8.9315 x 103
5.93 x 10 - 4
0.000 000 40
4.0 x 10 - 7
1 x 104
5.60 x 1011
1 x 10-5
5.02 x 10-8
0.000 000 0502
Multiplication and division.
Report your answer with the same number of digits as the quantity have the smallest number of significant figures.
Example. Density of a rectangular solid.
251.2 kg / [ (18.5 m) (2.351 m) (2.1m) ]
= 2.750274 kg/m3
= 2.8 kg / m3
(2.1 m - only has two significant figures)
An answer can’t have more significant figures than the quantities used to produce it.
How fast did the man run
if he went 11 km in
What is the Volume of this box?
Volume = length x width x height
= (18.5 m x 2.351 m x 2.1 m)
= 91.33635 m3
= 91 m3
(3.0 x 104) x (3.0 x 105) =
9.0 x 109
(6.0 x 105) x (2.0 x 104) =
12 x 109
But 12 x 109 =
1.2 x 1010
2.0 x 106
1.0 x 104
1.0 x 104
2.0 x 106
2.0 x 102
0.50 x 10-2
= 5.0 x 10-3
(2.3 x 104) + (4.1 x 104) =
6.4 x 104
*Exponent must be the same!*
(1.400 x 105) + (3.200 x 103) =
(140.0 x 103) + (3.200 x 103) =
143.2 x 103
1.432 x 105
After calculations, the last thing you do is round the number to correct number of significant figures.
If the first insignificant digit is 5 or more,
- you round up
If the first insignificant digit is 4 or less,
- you round down.
If a set of calculations gave you the following numbers and you knew each was supposed to have four significant figures then -
2.5795035 becomes 2.580
34.204221 becomes 34.20
1st insignificant digit
Many different systems for measuring the world around us have developed over the years.
People in the U.S. rely on the English System.
Scientists make use of SI units so that we all are speaking the same measurement language.
45 has little meaning, just a number
45 g has some meaning - mass
45 g /mL more meaning - density
Metric Units One base unit for each type of measurement. Use a prefix to change the size of unit.
Some common base units.
Mass gram g
Length meter m
Volume liter L
Time second s
Temperature Kelvin K
Energy joule J
Prefix Symbol Factor
giga G 109 1 000 000 000
mega M 106 1 000 000
kilo k 103 1 000
hecto h 102100
deca da 101 10
base - 100 1
deci d 10-1 0.1
centi c 10-2 0.01
milli m 10-3 0.001
micro or mc 10-6 0.000 001
nano n 10-9 0.000 000 001
Changing the prefix alters the size of a unit.
Powers of Ten http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html
Mass - the quantity of matter in an object.
Weight - the effect of gravity on an object.
Since the Earth’s gravity is relatively constant, we can interconvert between weight and mass.
The SI unit of mass is the kilogram (kg). However, in the lab, the gram (g) is more commonly used.
Units of measurement
Method of measurement
Quantity Definition Derived Unit
Area length x length m2
Volume length x length x length m3
density mass per unit volume kg/m3
speed distance per unit time m/s
acceleration speed per unit time m/s2
Force mass x acceleration kg m/s2 N
Pressure force per unit area kg/m s2 Pa
Energy force x distance kg m2 / s2 J
Volume - the amount of space that an object occupies.
Is a way of solving problems or answering questions.
Starts with observations and recording facts.
Hypothesis- an educated guess as to the cause of the problem or poses an answer to the question.
Experiment- designed to test the hypothesis
only two possible answers
hypothesis is right
hypothesis is wrong
Generates data observations from experiments.
Modify hypothesis - repeat the cycle
The hypothesis gets more and more certain.
Becomes a theory
A thoroughly tested model that explains why things behave a certain way.
Useful because they predict behavior
Help us form mental pictures of processes (models)
Scientific Law is developed
Description of how things behave
Law - how