Systems of Measurement

Systems of Measurement

Overview of Lecture Content

• Focus: Chapter three from math textbook focusing on systems of measurement.

• Key Topics:

  • United States Customary System

  • Metric System

  • Unit conversions within the metric system

  • Conversions between the metric system and the US system

  • Temperature scales: Celsius, Fahrenheit, and Kelvin

United States Customary System

• Context: The US is one of the few countries that predominantly use the United States customary system.

• Cultural Preference for Measurements:

  • Common units cherished by Americans: foot for hot dogs, gallons for milk and ice cream, miles for distances.

  • Example distance: "It's 52 miles to Sioux Falls."

• Inconsistencies with the US System:

  • Length measurements are inconsistent:

  • 1 foot = 12 inches

  • 3 feet = 1 yard

  • 5,280 feet = 1 mile

  • Volume measurements:

  • 1 teaspoon = 1/3 tablespoon

  • 8 fluid ounces = 1 cup

  • Mass measurements have four different systems:

  • Precious metals: 12 ounces = 1 pound

  • Pharmacy measurements: 12 ounces = 1 pound

  • Baby weight: 16 ounces = 1 pound

  • General purpose: 16 ounces = 1 pound

• Conclusion on US System: The overall inconsistency can lead to confusion in practical usage.

Development of the Metric System

• History: Developed in France during the 1790s to standardize measurements and minimize inconsistencies.

• Base of Metric System: Based on a scale of 10.

• Three Basic Units:

  • Length: Meter (m)

  • Definition: The length of the path traveled by light in a vacuum in 1/299,792,458 seconds.

  • Mass: Gram (g)

  • Definition: The mass of water contained in a cube having 0.1 meter sides (1 kg = 1,000 g).

  • Volume: Liter (L)

  • Definition: The volume of liquid contained within a cube of dimensions 0.1 meters.

Metric Prefixes and Conversions

• Addition of Prefixes: Used to determine unit size relative to the base unit:

  • Larger units: decimeter, hectometer, kilometer

  • Smaller units: decimeter, centimeter, millimeter

• Mnemonic for Conversion: "King Henry Died Unusually Drinking Chocolate Milk"

  • Stands for:

  • Kilo-

  • Hecto-

  • Deca-

  • Unit (meter, liter, gram)

  • Deci-

  • Centi-

  • Milli-

• Conversion Example: Converting between metric units through ratio and proportion, maintaining a base of 10 for easy calculations.

  • Example: Converting meters to millimeters:

  • Ratio: 1 meter = 1,000 mm → 3 meters = $3 imes 1000 = 3000$ mm

Unit Conversion Examples

• Problem: Convert 45 liters to milliliters.

  • Setup: 45 L / x mL = 1 L / 1,000 mL

  • Cross-multiply: $x = 45 imes 1000 = 45,000$ mL

• Problem: Convert 8,800 grams to kilograms.

  • Setup: 8,800 g / x kg = 1,000 g / 1 kg

  • Cross-multiply: $x = 8800 / 1000 = 8.8$ kg

• Problem: Convert 8 meters to centimeters.

  • Setup: 8 m / x cm = 1 m / 100 cm

  • Cross-multiply: $x = 8 imes 100 = 800$ cm

• Problem: Convert 5 hectoliters to deciliters.

  • Convert first to liters: 5 hL = 500 L → 500 L / x dL.

  • Continue: 1 L = 10 dL → $500 imes 10 = 5,000$ dL.

Alternative Conversion Approaches

• Decimal Place Movement Method: Based on the metric system's base of 10, moving the decimal place can simplify conversions:

  • Example:

  • From liters to milliliters, moving the decimal three places to the right for conversion.

• String Method for Conversions: Converting units by multiplying using equivalent ratios. Example for converting 4 hectograms to milligrams:

  • Setup:

  • 4Hg = x mg

  • Step 1: Convert Hg to g (1 Hg = 100 g)

  • Step 2: Convert g to mg (1 g = 1000 mg)

Inter-System Conversions: US and Metric

• Ratio and Proportions: Utilize conversion factors to convert between US and metric systems:

  • Example: Convert 150 pounds to kilograms using a conversion factor of 0.453.

  • Calculation:

  • $150 imes 0.453 = 67.95$ kg.

Temperature Scales

• Historical Development:

  • Daniel Gabriel Fahrenheit invented one scale, labeling freezing at 32 °F and boiling at 212 °F.

  • Anders Celsius developed another scale of 0 °C and 100 °C for freezing and boiling points, respectively.

  • Kelvin scale introduced in 1852, based on absolute zero, considered the official science measurement of temperature.

• Conversions:

  • Celsius to Kelvin: $K = C + 273.15$

  • Celsius to Fahrenheit: $F = (9/5)C + 32$ or $F = 1.8C + 32$

  • Fahrenheit to Celsius: $C = (F - 32) imes 5/9$

Examples of Temperature Conversions:

• Convert 77 °F to Celsius:

  • Calculation:

  • $C = (77 - 32) imes (5/9) = 25$ °C

• Convert 59 °C to Fahrenheit:

  • Calculation:

  • $F = (1.8 imes 59) + 32 = 138.2$ °F