Transcript Significant Figures - Miami Beach Senior High School

AIM: Significant Figures
►What are
significant
figures?
►On a blank sheet of paper
Chapter Two
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AIM: Significant Figures
►The number of significant figures is the
number of digits believed to be correct
by the person doing the measuring. It
includes one estimated digit
Chapter Two
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Lab Report
Title page
Title page is present
and contains the
student's name, class,
date, teacher's name
and title of the lab.
Members of your
group and boxes
tested
Title page is present
but is missing one or
more of the following
components:
student's name, class,
date, teacher's name,
members of your
group and title of the
lab.
No title page provided
Problem
Inquiry problem is
clearly stated.
Inquiry problem is not
clearly stated.
No inquiry problem
provided.
Experimental
Hypothesis
Hypothesis for each
box provides an
educated guess that is
clear and reasonable
based on what has
been studied.
Hypothesis provides
an educated guess but
is not logical.
No hypothesis has
been stated.
Materials
All materials and
setup used in the
experiment are clearly
and accurately
described.
Almost all materials
and the setup used in
the experiment are
clearly and accurately
described.
Many materials are
described inaccurately
OR are not described
at all.
3
Lab Report
Procedures
Procedures are
listed in clear steps.
Each step is
numbered and is a
complete sentence.
Procedures are
listed in a logical
order, but steps are
not numbered
and/or are not in
complete sentences.
Procedures do not
accurately list the
steps of the
experiment.
Results
Results give a
detailed description
of what happened
when you performed
your tests for each
box. You may include
tables that
summarize your
results for each box
tested
Results give some
description of what
happened when you
performed your
tests.
No results provided.
Conclusion
Conclusion includes
what was predicted
to be inside the
boxes and how
whether this
prediction was
correct or not.
Conclusion includes
what was predicted
to be inside the
boxes but does not
indicate if this
prediction was
correct or not.
No conclusion was
included in the
report OR shows
little effort and
reflection.
4
Significant Figures
► When using our calculators we must determine the correct
answer; our calculators are mindless drones and don’t know
the correct answer.
► There are 2 different types of numbers
– Exact
– Measured
► Exact numbers are infinitely important
► Measured number = they are measured with a measuring
device so these numbers have ERROR.
► When you use your calculator your answer can only be as
accurate as your worst measurement…
Chapter Two
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Exact Numbers
An exact number is obtained when you count objects
or use a defined relationship.
Counting objects are always exact
2 soccer balls
4 pizzas
Exact relationships, predefined values, not measured
1 foot = 12 inches
1 meter = 100 cm
For instance is 1 foot = 12.000000000001 inches? No
1 ft is EXACTLY 12 inches.
6
Learning Check 1
A. Exact numbers are obtained by
1. using a measuring tool
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
2. counting
3. definition
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Solution
A. Exact numbers are obtained by
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
8
Learning Check 2
Classify each of the following as an exact or a
measured number.
1 yard = 3 feet
The diameter of a world cup soccer ball is 6 x 106
cm.
There are 6 hats on the shelf.
Gold melts at 1064°C.
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Solution
Classify each of the following as an exact (1) or a
measured(2) number.
This is a defined relationship.
A measuring tool is used to determine the value.
The number of hats is obtained by counting.
A measuring tool is required.
10
2.4 Measurement and Significant
Figures
► Every experimental
measurement has a
degree of uncertainty.
► The volume, V, at right
is certain in the 10’s
place, 10mL<V<20mL
► The 1’s digit is also
certain, 17mL<V<18mL
► A best guess is needed
for the tenths place.
Chapter Two
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What is the Length?
►We can see the markings between 1.6-1.7cm
►We can’t see the markings between the .6-.7
►We must guess between .6 & .7
►We record 1.67 cm as our measurement
►The last digit an 7 was our guess...stop there
12
Learning Check 3
What is the length of the wooden stick?
1) 4.5 cm
2) 4.54 cm
3) 4.547 cm
? 8.00 cm or 3 (2.2/8)
14
Measured Numbers
►Do you see why Measured Numbers have error…you
have to make that Guess!
► All but one of the significant figures are known with
certainty. The last significant figure is only the best
possible estimate.
► To indicate the precision of a measurement, the
value recorded should use all the digits known with
certainty.
15
Below are two measurements of the mass of the
same object. The same quantity is being described
at two different levels of precision or certainty.
Chapter Two
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Note the 4 rules
When reading a measured value, all nonzero digits
should be counted as significant. There is a set of
rules for determining if a zero in a measurement is
significant or not.
► RULE 1. Zeros in the middle of a number are like any
other digit; they are always significant. Thus, 94.072
g has five significant figures.
► RULE 2. Zeros at the beginning of a number are not
significant; they act only to locate the decimal point.
Thus, 0.0834 cm has three significant figures, and
0.02907 mL has four.
Chapter Two
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► RULE 3. Zeros at the end of a number and after
the decimal point are significant. It is assumed
that these zeros would not be shown unless they
were significant. 138.200 m has six significant
figures. If the value were known to only four
significant figures, we would write 138.2 m.
► RULE 4. Zeros at the end of a number and before
an implied decimal point may or may not be
significant. We cannot tell whether they are part
of the measurement or whether they act only to
locate the unwritten but implied decimal point.
Chapter Two
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Practice Rule #1 Zeros
45.8736
6
•All digits count
.000239
3
•Leading 0’s don’t
.00023900 5
•Trailing 0’s do
48000.
5
•0’s count in decimal form
48000
2
•0’s don’t count w/o decimal
3.982106 4
1.00040
6
•All digits count
•0’s between digits count as well
as trailing in decimal form
2.5 Scientific Notation
► Scientific notation is a convenient way to
write a very small or a very large number.
► Numbers are written as a product of a number
between 1 and 10, times the number 10
raised to power.
► 215 is written in scientific notation as:
215 = 2.15 x 100 = 2.15 x (10 x 10) = 2.15 x 102
Chapter Two
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Two examples of converting standard notation to
scientific notation are shown below.
Chapter Two
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Two examples of converting scientific notation back to
standard notation are shown below.
Chapter Two
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► Scientific notation is helpful for indicating how
many significant figures are present in a number
that has zeros at the end but to the left of a decimal
point.
► The distance from the Earth to the Sun is
150,000,000 km. Written in standard notation this
number could have anywhere from 2 to 9 significant
figures.
► Scientific notation can indicate how many digits are
significant. Writing 150,000,000 as 1.5 x 108
indicates 2 and writing it as 1.500 x 108 indicates 4.
► Scientific notation can make doing arithmetic easier.
Rules for doing arithmetic with numbers written in
scientific notation are reviewed in Appendix A.
Chapter Two
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2.6 Rounding Off Numbers
► Often when doing arithmetic on a pocket
calculator, the answer is displayed with more
significant figures than are really justified.
► How do you decide how many digits to keep?
► Simple rules exist to tell you how.
Chapter Two
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► Once you decide how many digits to retain, the
rules for rounding off numbers are straightforward:
► RULE 1. If the first digit you remove is 4 or less, drop
it and all following digits. 2.4271 becomes 2.4 when
rounded off to two significant figures because the
first dropped digit (a 2) is 4 or less.
► RULE 2. If the first digit removed is 5 or greater,
round up by adding 1 to the last digit kept. 4.5832 is
4.6 when rounded off to 2 significant figures since
the first dropped digit (an 8) is 5 or greater.
► If a calculation has several steps, it is best to round
off at the end.
Chapter Two
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Practice Rule #2 Rounding
Make the following into a 3 Sig Fig number
1.5587
1.56
.0037421
.00374
1367
1370
128,522
129,000
1.6683 106
1.67 106
Your Final number
must be of the same
value as the number
you started with,
129,000 and not 129
Examples of Rounding
For example you want a 4 Sig Fig number
0 is dropped, it is <5
4965.03
4965
780,582
780,600 8 is dropped, it is >5; Note you
must include the 0’s
1999.5
2000.
5 is dropped it is = 5; note you
need a 4 Sig Fig
RULE 1. In carrying out a multiplication or division,
the answer cannot have more significant figures than
either of the original numbers.
Chapter Two
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►RULE 2. In carrying out an addition or
subtraction, the answer cannot have more digits
after the decimal point than either of the
original numbers.
Chapter Two
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Multiplication and division
32.27  1.54 = 49.6958
49.7
3.68  .07925 = 46.4353312
46.4
1.750  .0342000 = 0.05985
.05985
3.2650106  4.858 = 1.586137  107
1.586 107
6.0221023  1.66110-24 = 1.000000
1.000
Addition/Subtraction
25.5
+34.270
59.770
59.8
32.72
- 0.0049
32.7151
32.72
320
+ 12.5
332.5
330
Addition and Subtraction
.56
__ + .153
___ = .713
__ .71
82000 + 5.32 = 82005.32
82005
10.0 - 9.8742 = .12580
.1
10 – 9.8742 = .12580
0
Look for the
last important
digit
Mixed Order of Operation
8.52 + 4.1586  18.73 + 153.2 =
= 8.52 + 77.89 + 153.2 = 239.61 =
239.6
(8.52 + 4.1586)  (18.73 + 153.2) =
= 12.68  171.9 = 2179.692 =
2180.
Homework Due 10/09/2013
►Use your calculator to do problems 16-19 on the
practice problems worksheet
Chapter Two
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Practice problems
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