Chapter 11: Measurement

Section 11.1: Fundamentals of Measurement

See Activity 11A regarding Measurable Attributes

Structure of Measures

• Create an order (taller vs shorter, heavier vs lighter)

• Some are additive• Ex’s: length, weight, time• Non-Ex’s: temperature, density

Definitions

•Def: unit- a reference amount for a given quantity

•Def: To measure a quantity- to compare that quantity with a unit of the quantity

UnitsWhat makes a particular measurement a good choice as a unit?• Precise, standardized definition• Easy to convert to other units• Easily measurable

Some bad units:• Steps or paces• Square miles if measuring the size of a room

Another bad unit

• Lordes

Standard U.S. System Units

Deciding on which unit to use

Ex: Which unit would you use to measure the following?a) The area of this classroomb) The weight of a pencilc) The length of a basketball courtd) The volume of a water bottlee) The volume of a textbook

The Units for the Metric System

Metric System Prefixes

Definitions of metric units

•meter- originally defined as the length of a pendulum whose half-period is exactly 1 second, or as 1/10,000,000 of the distance from the equator to the North Pole

• cc- cubic centimeter, used for dosages

•metric ton- 1000 kilograms

Important relationships among metric units

•1 milliliter of water weighs 1 gram• No easy comparison for number of ounces a fluid ounce of water

weighs

•1 milliliter of liquid fills a 1 cubic centimeter container• No easy comparison for fluid ounces and cubic inches

Converting between U.S. and metric systems• Length: 1 inch = 2.54 cm

• Capacity: 1 gallon ≈ 3.79 liters

• Weight/mass: 1 kg ≈ 2.2 pounds

Measuring Length

• How do we measure length?Line units up end to end and count how many units make upthe quantity’s length

• What if the object isn’t straight?Run a string along the object or surface then measure the string’s length when it is straightened out

Section 11.2: Length, Area, Volume, and Dimension

Length

• Def: Length- describes the size of something that is one-dimensional- equal to the number of a chosen unit of length that it takes to cover the object without gaps or overlaps

• Def: one-dimensional- an object where at each point, there is only one independent direction to move within the object• Ex’s: line segment

outer edge of a circle or polygon curved line

equator of the Earth imaginary line from one end of a 2 or 3-dimensional object to the other

end

Perimeter• Def: The perimeter is the “distance around a shape,” or the length of

the outer edge of a shape.• Ex 1: How many feet of baseboard do you need to go around the

following room?

Assume each square in the grid is 1 square foot.

Perimeter• Def: The perimeter is the “distance around a shape,” or the length of

the outer edge of a shape.• Ex 1: How many feet of baseboard do you need to go around the

following room?

Area• Def: Area- describes the size of a two-dimensional object

- equal to the number of a chosen unit of area that it takes to cover the object without gaps or overlaps

• Def: 2-dimensional- at each point, there are 2 independent directions to go within the object • Ex’s: coordinate plane

inside of a circle or polygon piece of paper

surface of a box land surface of the Earth

Example Area Problem

• Ex 2: How many square feet of carpet do you need to cover the floor of the following room?

Example Area Problem

• Ex 2: How many square feet of carpet do you need to cover the floor of the following room?

Volume

• Def: Volume- describes the size of a 3-dimensional object - equal to the number of a chosen unit of volume that it takes to fill the object without gaps or overlap

• Def: 3-dimensional- at each point, there are 3 independent directions to go within the object• Ex’s: atmosphere

inside of a balloon or box water in a container the Earth

Example Volume Problem

• Ex 3: How many cubic feet of air is in the following room if the flat ceiling is 10 feet from the floor?

Section 11.3: Error and Precision in Measurements

Example Error Problem

• Ex 1: If a car company lists its sedan as weighing 3110 lbs, what could the car’s actual weight be?

Another Example Problem

• Ex 2: If Colorado is reported to be 380 miles wide and 280 miles long, how large can the error be if the area is reported as 106,400 square miles?

Section 11.4: Converting Between

Units of Measurement

Converting Units

• Ex 1: A scale measures a dog’s weight to be 356 ounces. How many pounds does the dog weigh?

(Recall that )

Converting Units

• Def: Dimensional Analysis: process for converting between units by multiplying the number 1 expressed as a fraction that relates two units

• Ex 2: How many miles are in 5 kilometers? (Recall that )

See practice problems on Activity 11G

Converting Units of Area and Volume

• Ex 3: Paulene is making a quilt which is made with patterned squares of size How many square pieces does she need to sew together to make the quilt?

See more practice problems in Activity 11I

Approximate Conversions

Using Approximate Conversions

• Ex 4: Gas in England is currently and in the U.S. it is per gallon. In which country is gas cheaper?