Welcome to Math 107 – Math in Society
1. Reflect for a moment on what life must have been like for early
human hunter-gatherers. In groups, discuss the cool things in your
life that make this a better time to be alive. We will compile your
ideas on the board.
2. Identify some problems our society has created for itself in pursuit
of making life better that were not problems for the huntergatherers. We will compile your ideas on the board.
3. Discuss with your group whether you think the successes of
humanity will lead to a better world in the future or the problems we
have created will lead to a worse future.
Goals are important and while it is difficult to get consensus of
all the world’s people, I am proposing that mankind seek a vibrant
but sustainable future.
What I mean by vibrant is: good health, sports, learning, inventing,
creating, exploration, art, theater, music, singing, dancing, good food,
socializing, debating, nature, contemplation, etc.
Anti-vibrant includes authoritarianism, discrimination, injustice,
violence, and war.
Sustainable means that our pursuit of a vibrant life does not
eliminate this option for future generations.
Maslow’s Hierarchy of Needs
http://www.simplypsychology.org/maslow.html
Solving problems requires asking the right questions and then using
the appropriate strategy to answer them. For example,
1. How "sacred" is life itself? How heroic are we obligated to be in
near-terminal conditions? Should we begin to consider rationing
health care for the very old, those with senility?
2. What is the best way to pace a race?
3. How much money will you have in an account in 20 years if you
put $50 in the account every month?
4. What proportion of people considers sports to be important in a
vibrant society?
This class will draw on your prior knowledge of arithmetic and
algebra. When working with numbers, it is important to consider
units. The table below contains a variety of units. We will discuss
these categories as they will play a role in the remainder of this
course.
USCS (US
Customary
System)
Metric or SI
USCS (US Customary
System)
Mass
Length
inches (in)
feet (ft)
yard (yd)
mile (mi)
square inches
(in2)
square feet (ft2)
square yards (yd2)
square mile (mi2)
acres (a)
ounces (oz)
cups (c)
pints (pt)
quarts (qt)
gallon (gal)
cubic inches (in3)
cubic foot (ft3)
millimeters (mm)
centimeters (cm)
meter (m)
kilometer (km)
Area
square centimeters (cm2)
square meters (m2)
square kilometers (km2)
hectares (h)
Volume
milliliters (ml)
liters (L)
cubic centimeters (cm3)
cubic meter (m3)
Metric or SI
ounces (oz)
pounds (lb)
ton (t)
milligrams (mg)
grams (g)
kilogram (kg)
metric ton (MT)
Power, Energy and Work
joules (J)
watts (W)
kilojoule (kJ)
kilowatt (kW)
calories (cal)
kilocalorie (kcal)
kilowatt hour (kWh)
British Thermal Units
(BTU)
megajoule (MJ)
Time
Seconds
Minutes
Hours
Days
Dimensional Analysis - A formal process for converting from one set of units to
another. Unit equivalencies are necessary for the conversions.
USCS (US Customary System)
Unit Equivalencies
USCS – Metric
Metric or SI
Length
12 inches (in) = 1 foot (ft)
3 feet (ft) = 1 yard (yd)
1760 yards (yd) = 1 mile (mi)
5280 feet (ft) = 1 mile (mi)
1 square mile (mi2)= 640 acre
1 acre = 43,560 square feet (ft2)
8 ounces (oz) = 1 cup (c)
2 cups (c) = 1 pint (pt)
2 pints (pt) = 1 quart (qt)
4 quarts (qt) = 1 gallon (gal)
1 cubic foot (ft3)=7.481 gallons (gal)
16 ounces (oz) = 1 pounds (lb)
2000 pounds (lb) = 1 ton
1000 Watts = 1 kilowatt
1000 calories (cal) = 1 kilocalorie
(kcal) = 1 Calorie (Cal)
1 kilowatt hour (kWh)= 3412 British
Thermal Units (BTU)
1 inch (in) = 2.54 centimeters (cm)
0.62 miles (mi) = 1 kilometer (km)
Area
2.471 acre = 1 hectare
1 square mile (mi2) = 2.59 square
kilometers (km2)
Volume
1 quart (qt) = 0.946 liters (L)
1000 millimeters (mm) = 1 meter (m)
1000 meters (m) = 1 kilometer (km)
100 centimeters (cm) = 1 meter (m)
1 square kilometer (km2) = 100 hectare
1 hectare = 10,000 square meters (m2)
1000 milliliters (ml) = 1 liter (L)
1000 liters (L) = 1 cubic meter (m3)
Mass
2.20 pounds (lb) = 1 kilogram (kg)
1 pound (lb) = 453.6 grams (g)
Power, Energy and Work
1 calorie (cal) = 4.187 Joules (J)
1 Watt-Second (W·S) = 1 Joule (J)
Time
60 Seconds (s) = 1 minute (min)
60 minutes (min) = 1 hour (h)
24 hours (h) = 1 day (d)
365 days (d) = 1 year (y)
1000 milligrams (mg) – 1 gram (g)
1000 grams (g) = 1 kilogram (kg)
1000 kilograms = 1 metric ton
1 kilojoule (kJ)= 1000 joules (J)
1 megajoule (MJ) = 1,000,000 joules(J)
As an example of the importance of units, would you take a job that offered to pay you
100?
Without units, we don’t know if that is
100 cents
100 dollars.
We also don’t know the rate,
100 cents per day
100 dollars per hour
100 yen per hour
The units of choice are dependent upon the cultural preference of the people or
organization. Sometimes it is necessary to convert from once set of units to another.
The best way to do this is with dimensional analysis.
Convert 100 Euros to dollars if 1 dollar = 0.82 Euros.
Conversions often take place with linear, area, and volume measurements.
Linear: Convert 25 yards to feet
Area: Convert 5 square yard to square feet
Volume: Convert 4 cubic yards to cubic feet
More than one unit fraction can be used when doing dimensional analysis.
Convert 1 day into seconds:
Convert 1,000,000 seconds into hours.
Convert rates such as 60 miles per hour to feet per second
Students:
The Tacoma Narrows Bridge is 1822 meters. Convert this distance to miles.
King County Aquatics Center was built in 1990 for the Goodwill Games. The
main competitive pool measures 50 m x 25 yd by 9.5 ft. How many gallons of
water does it hold?
My house has about 15 light bulbs. If they are all 60 watt bulbs that are used
for an average of 3 hours a day, how much money is spent for lighting my
house if electricity costs $0.07 per kWh?
How do you determine the fuel economy of your car?
Assign: Experiential 1 – Fuel Economy, due in 2 weeks.
Activity 1.1 Dimensional Analysis
Formulas
Geometric formulas – areas, perimeters, volumes
Physics formulas
Statistical formulas
Financial/economic formulas
Formulas show the relationship between variables. For example, the relationship
between the distance a person travels, their speed, and the amount of time they are
traveling is d=rt.
Formulas have been developed by researchers and mathematicians over the centuries.
Using a formula that already exists is easier than finding the relationships on your own.
To use a formula, follow the 3-step process of writing the formula, showing the
substitution, and then calculating.
Some examples of formulas you will use in this class include:
pˆ  2
Pt = P0(1+r)
Pt = P0ert
t
pˆ 1  pˆ 
n
r

Pt  P0 1  
k

APY = er-1
x2
s
n
tk

APY  1 

k
r
 1
k
12t


r 
d 1    1
 12 

Pt  
r
12
Similar to problems in the draft 4 of the book
Example. The formula pˆ  2
pˆ 1  pˆ 
is used for determining a confidence interval. Perhaps you have heard poll
n
results just prior to an election in which they talk about a margin of error. They used this formula to get their
result. In this formula, is the proportion of their sample, n is the size of their sample. If the sample proportion is
43% and the sample size is 460, what is the confidence interval?
Formula:
Substitution:
Solution:
Example. The formula
is used to determine the amount of a monthly payment for repaying a
loan. P0 is the amount of money that is borrowed, r is the interest rate, t is the number of years. What is the
monthly payment of a $50,000 loan with a 4.1% interest rate if the term is 8 years?
Formula:
Substitution:
Solution:
Students: The formula
is used to determine the amount of money in an account in year t, with
an initial investment of P0, a rate of r, k compounding periods in a year. How much will be in an account if $3000
is invested in an account that has an interest rate of 2.3%, compounded quarterly (k = 4).
Formula:
Substitution:
Solution:
Activity 1.2
Proportional Reasoning
A proportion is the ratio of the part to the whole. It can be expressed as a
decimal, percent, or per some other number such as 10,000 or 1 million.
According to Gallup, out of 946 people with investments for retirement, 246
began their retirement investments before age 25. (This is a topic we will
discuss in Chapter 4) http://www.gallup.com/poll/180341/one-four-investors-started-young-saving-retirement.aspx
What proportion of people with retirement investments began before age 25?
What percent of people with retirement investments began before age 25?
A percent is a proportion multiplied by 100. Percents are equivalent to the
concept of per 100.
Sometimes, especially when the proportion is small, these numbers are based
on larger numbers than 100. For example,
What is the number of people with retirement investments they began before
age 25 per 100,000?
Or 26,000 per 100,000
Students:
In 2011, an estimated 13,834 people died of AIDS. There were 312 million
people in the US in 2011.
Find the proportion of people who died.
Find the percent of people who died.
Find the number of people per 10,000 who died.
Find the number of people per 100,000 who died.
Find the number of people per 1 million who died.
The number of people who get cancer is called the cancer incident rate. In
WA, the cancer incident rate is 460.1 per 100,000. If WA has 7.1 million
people, how many people in the state can expect to get cancer this year?
http://www.cdc.gov/cancer/dcpc/data/state.htm
The cancer incident rate in Oregon is 422 per 100,000. If they had 16,761
cases of cancer last year, what is their population?
Population Density
Another type of proportion is population density. It indicates the
amount of space each person would have if everyone had the same space.
For example, Lakewood has 60,000 people in an area of 20 square miles. How
many people does it have per acre?
Do you think Seattle has a larger or smaller population density than
Lakewood?
Seattle has a population of 3,600,000 and an area of 5,872 square miles.
What is Seattle’s population density in people per acre?
Comparisons
Comparisons between numbers can be made in absolute terms or relative
terms using percents. Care must be taken in the phrasing of the results.
For example:
Lakewood has 3,540,000 fewer people than Seattle.
Seattle has 3,540,000 more people than Lakewood.
Lakewood is 1.67% of the size of Seattle (60,000/3,600,000)
Lakewood is 98.3% smaller than Seattle (3,540,000/3,600,000)
Seattle is 60 times larger than Lakewood (3,600,000/60,000)
It is not correct to say that Lakewood is 60 times smaller than Seattle!
Experiential 2 – What is the average amount of water this is used when taking
a shower?