1. Measuring and Calculating
Chapter 3

2. Outline Measuring and Units Measurements Mathematics and Measurements
Orderly Problem Solving

3. What is Measuring?
Measurement is the act of comparing an unknown quantity to a standard unit
A dimension is a frame of measurement
Length, mass, volume, time, and electrical charge are a few examples
A unit has to be defined in order for something to be measured
A measurement consists of two parts:
A number
A unit in which the measurement was taken. (Only like units can be added/subtracted but any units can be multiplied or divided when needed)

4. Metric Systems of Measurement
The metric system is a measuring system based on a decimal scale, which just means the base units are by power of tens
The widely accepted metric scale is called the International System of Units or SI
In the 18th Century a need for a more standardized scale spurred the birth of the metric scale

5. SI Units Dimension Dimensional symbol SI Unit Unit Symbol Length 𝑙
There are 7 base units in the SI which covers basic dimensions:
Dimension
Dimensional symbol
SI Unit
Unit Symbol
Length 𝑙
Meter m
Mass M
Kilogram kg
Time t
Second s
Temperature T
Kelvin K
Number of particles n
Mole
mol
Electrical Current l
Ampere A
Light Intensity IL
Candela cd
Look at the tables on page 50 and 51

6. SI Unit Prefixes
The SI allows scientists to size units so that they are convenient to use
In order to change the size of the units, SI adds prefixes that represent powers of 10 to make the unit either larger or smaller
KNOW THE TABLE ON PAGE 52

7. Conversion Factors
Unit conversion consists of multiplying the measurement by a conversion factor
A conversion factor is a fraction that contains both the original unit and its equivalent value in a new unit
Example: Convert liters to milliliters
I L= 1000 mL

8. Bridge Notation
Bridge Notation is a special notation for multiplying and dividing several measurements together at the same time.
Example: How many seconds are in 1 year

9. More Examples Convert the following using bridge notation:
cm to kilometers
350 mg to kilograms

10. Accuracy
There will always be uncertainty in measurements as long as humans are the ones taking measurements
The accuracy of a measurement is a numerical evaluation of how close the measured value is to the actual or accepted value of the dimension measured
Whenever you measure something, you usually take a measurement knowing what it is supposed to be. By knowing the “accepted value” you can see how far off you are by calculating the percent error
Percent error = observed value −accepted value accepted value ∗100%
Example on page 56

11. Precision
Precision indicates how repeatable a measurement is or how exactly one can make a measurement
Precision is counting something 5 different times and getting the same number every single time….however, it doesn’t mean that what you counted was right
Accuracy and precision are not the same thing!

13. Significant Digits
When taking a measurement, the last digit is always the uncertain or least significant digit because when we take a reading we are often the most unsure about where that measurement falls
The significant digits in a measurement are only those that are known for certain plus one estimated digit
When reading a thermometer you always say “wellllll it could be .1 or maybe .0

14. Significant Digits Rules
Significant digits apply only to measured data
SD do not apply to counted or pure numbers
SD do not apply to ratios that are exactly 1 by definition
All nonzero digits are significant
25.4 mL = 3 SDs
13.78 g = 4 SDs
All zeros between nonzero digits are significant
1003 cm = 4 SDs
1.09 g = 3 SDs
Pure numbers are those without units and counted numbers are those that are exact 5 balloons

15. Significant Digit Rules
Decimal points define significant zeros
If a decimal point is present, all zeros to the right of the last nonzero digit are significant
20.0 s = 3 SDs
= 5 SDs
If a decimal point is not present, no trailing zeros are significant
2500 mL = 2 SDs
In decimal numbers, all zeros to the left of the first nonzero digit are not significant. They are placeholders
0.075 kg = 2 SDs
s = 2 SDs

16. Significant Digit Rules
Significant zeros in the one’s place are followed by a decimal point
2500. mL = 4 SDs
The decimal factor of scientific notation contains only significant digits
2.50 x 103 mL = 3 SDs

17. Significant Digit Examples
For each of the following determine the number of SDs
9.370 kg g mL
12 cookies s

18. Calculations with Measured Data
Measured data must be the same kind of dimension and have the same units before that can be added or subtracted
The sum or difference of measured data cannot have greater precision than the least precise quantity in the sum or difference
Example: Add m, m, and 23.7 m

19. Calculations with Measured Data
The product or quotient of measured data cannot have more SDs than the quantity with the fewest SDs
Example: What is the area of a piece of paper that is 11.5 cm ling and 5.5 cm wide?
The product or quotient of a measurement and a counted number, conversion factor, or defined value has the same number of decimal places, or same precision, as the original measurement
3 x 3.64 g = g

20. Calculations with Significant Digits
In compound calculations, do not round off at the intermediate steps

21. Steps in Problem Solving
Read the problem
Read through the problem to figure out what is being asked and what data you are given
Determine the method of solution
Look for key words to show you how to solve the problem
Choose the specific tools to use
Figure out what type of equations you will have to use
Set up the problem, estimate, and calculate
Write down the equation, plug in your values, cancel units, and start calculating
Check and format
Check to make sure you really answered the question. Once you’re sure, make sure there are the right amount of SDs and make sure the unit is present

22. Example in Problem Solving
A lead fishing sinker has a mass of 51.0 g. If the density of lead is g/cm3, what is the volume of the sinker?