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The Math Behind Insurance.

  I work in a place where most people have government issued medical insurance, no personal insurance, no car insurance.  I don't even think they have insurance on their houses.

So when I talk about insurance, my students do not relate to it.  I admit, the only reason I knew about insurance before I graduated from high school was thanks to a class I took as a senior.

In that class, they had a variety of people from car sales to home sales, and so many more but they did make sure an insurance agent came in to talk to us about house, car, and all other types they sold.

The first thing insurance companies rely on is the law of large numbers. An example of this is when flipping a coin. On the first flip there is a 50 percent chance of getting a heads.  On the second flip, the chance of getting two heads in a row is 1/2 * 1/2 or 1/4 = 25%.  For the third flip, getting three heads in a row is 1/2 * 1/2 * 1/2 is 1/8 or 12.5%.  As the number of flips increases the chance of getting heads all in a row decreases such as flipping 6 heads in a row is 1/64 or 1.5%.

If instead you check for the percent of heads versus tails you get, the more times you flip the coin, the closer the percent gets to 50%.   Insurance companies keep track of each event such as car crashes, tornadoes taking out houses, etc and the larger the sample the more accurate the mathematical probability.

The second concept they use is one of "weighted probability" which takes into account everything.  If you played a dice game with a man where you would get $6 for every 6 you roll but you'd have to pay him $2 for any other number,  is it worth playing? 

You know the chance of rolling a 6 is 1/6 while the chance of rolling any other number is 5/6.  To calculate the weighted probability, its (-2)(5/6) + (6)(1/6) = -.66 or you would lose 66 cents each game. 

The idea is that more people will have nothing happen than those who have something happen.  So if you were a small insurance company with 1000 clients.  Say 1 house catches fires each year so the probability is 1/1000 of that happening.  Therefore the chances of the house not catching fire is 999/1000.  The replacement cost of the house is $200,000.  As far as premiums, each person pays $20 per month for a total of $240 per year.

The mathematics would be -200,000(1/1000) + 240(999/1000) = -200 + 239.76 = a profit of $39.76 per person.  This means your company will make a profit of $39,760 based on 1000 x $39.76.

This is a simple example but it gives a better idea of how insurance companies work.  Hope you find this interesting.  Let me know what you think, I'd love to hear.

Math and the Internet.

  When I was in Denver attending a conference, someone commented the Internet has undergone exponential growth.  Exponential?  I can almost believe the claim but think about letting students explore that to figure out if it is true.

This site has a wonderful chart describing the growth of the internet from December 1995 to December 2017.  It provides information on number of people who used the internet and the percent of the world population.

This would be a perfect thing to do a project on.  Students could create an excel spreadsheet to show the growth, determine the percent increase each year, figure out if the numbers justify the claim of exponential growth.  Students could even calculate the world population and its increase.  Just a couple of pieces of information and lots of fun things to calculate.

This site offers a 16 slide presentation showing a wonderful breakdown of information of who was using the internet in January 2012 from various geographic regions for the internet, social, and mobile uses.  This would be wonderful again for additional charts showing world wide uses by topic and geographic regions or provide a breakdown for the world based on combining all of the information.


What about letting students learn to read interactive charts.  This interactive site has an interactive chart showing the growth of internet uses by each geographic region, the current breakdown of internet uses by country, cell phone users world wide, and broadband penetration.  Several have the information displayed by chart and/or map, provides downloadable data files, and lists information sources.

If you'd prefer to have students compare domains, this site has two charts with the numbers. By comparing and calculating the growth over the years, students get a different perspective. 

It is easy to have students use the sites to create different graphs to compare the information in different forms before explaining which graph they believe is best to provide the information.  Not every type of graph can be used for every type of information.

If students choose one of the first three sites to create a report on the growth of the internet.  In the report, they should include the graphs which could show the growth itself, or percent growth and then explain what they see.  If the data indicates exponential growth, they could create an equation to fit the data.  They could also predict the numbers of users in 5, 10, or 15 years based on current growth trends.

This is applied real world math which requires students to analyze data they are given which is what mathematicians do in real life.

Let me know what you think, I'd love to hear.  

Ferris Wheel Math

  Just about any traveling amusement entertainment or amusement park has at least one Ferris wheel.  I remember living in the middle of a flat dusty part of New Mexico over near the Texas boarder and the carnival arrived in town with a large number of rides including the Ferris wheel.

I am the member of the family who hated going up on those rides because I had a horrible fear of heights and I still do.  I end up gripping the bar, closing my eyes, and praying till the ride is over.  I love looking at them from a distance and the math is so elegant but please don't make me get up in one.

Fortunately for me, there is a nice amount of math associated with a Ferris Wheel.  As you can tell from the weekend warm-ups, there is always the circumference and area.  Students could also design a scale model of a Ferris wheel complete with seats and everything.  They could determine how far apart the seats are either in feet or in degrees since the wheel has 360 degrees.

A Ferris Wheel has quite a lot of trig associated with it so its possible to use this topic in Geometry, Trigonometry, or Algebra II. For Geometry, you can calculate circumference, area,  and surface area.  The Ferris Wheel provides a great way of applying sine and cosine functions.  For Trig and Algebra II student can calculate rates for the Ferris Wheel

 If you have students create a Ferris Wheel out of paper and a brad, they can play with it to determine the height of a passenger car from the ground as it turns.  Students can take readings every 15 degrees.  Once done, the heights can be placed on  a graph so students are able to see the graph resembles a sine wave.

Students can also relate the unit circle to sine and cosine waves if the student places the sine and cosign values as riders in each car.  As the car hits the bottom where people get off, the values can be graphed showing the relationship between the unit circle and the graphs of both the sine and cosine.  

Another activity would be to place a circle on graph paper to determine which parts of the ride would have positive values, negative values, a mixture in respect to the x and y axis.  Let the student know they begin at the positive x axis.

In addition, its possible to include the math an engineer or designer might use to design a Ferris wheel. This site has a lovely write up on what parts are used to create one.  It is good to relate the application to the theory so students see practical applications for the math they are learning.  Its only due to teaching that I've found real life applications for much of the theory I'd learned at school. I love that but wish they'd covered it when I was in school.

Let me know what you think.  I'd love to hear.  Have a great day.

JPL Space Math and Pi

Yesterday, I introduced everyone to the fifth activity in a group of activities created by JPL called "Solar Sleuth: A "Pi in the Sky" Math Challenge!"

Yesterday I shared the fifth activity so today I'll share a bit more about the first four activities. The first one or beginning one is just labeled "Pi in the Sky". 

Its infographic introduces students to the Soil Moisture Active Passive or SMAP satellite, the Curiosity Mars rover, Juno orbiting Jupiter, and something on the Cassini spacecraft.  At the end of each explanation students will find a mathematical question to answer."Pi in the Sky 2" infographic introduces students to the Mars Exploration rover, the Dawn spacecraft and Ceres, Europa and a possible liquid ocean, and the twin Voyagers.  Each topic has a wonderful description ending with a question.

 "Pi in the Sky 3" follows the same format but covers Titan's atmosphere, the Mars Reconnaissance Orbiter,  the explanation of a transit, and the Juno spacecraft having to brake.  The last one, "Pi in the Sky 4" addresses impact craters, the 2017 eclipse, Cassini's death, and the hunt for a habitable planets.

What is coolest about all of these besides providing the infographic, student handouts, and answer sheets, is they provide a challenge slide show which puts all of the problems together into a slide show if you'd prefer to do it that way.  There is a worksheet for each slide.

There is also something explaining five ways NASA uses Pi and provides a problem for students to solve.  In addition, there is a link to a blog entry discussing the how many digit's of pi we really need when using it in calculations.

I love this set of activities because students are able to read infographics, identify the information needed to answer the question at the bottom.  Real world skills that are covered more often in science classes than math classes.  I plan to use these activities in math class over the year to give my students more chance to practice reading for information.

Let me know what you think, I'd love to hear.  Have a great day.

Space + Math

  I just spent 4 days at a conference with so many panels on space exploration.  There were talks on the sun, the solar system, space exploration, and so many other topics.

Many of the talks addressed many of the discoveries based on data sent back by Voyager, New Horizons, Kepler, and Cassini.  What I find so amazing is the mathematics involved in sending them out into space on a path to get them exactly where they want them many years in the future.

For the Voyager, the engineers had to keep in mind gravitational forces between the earth, the sun, the moon, the planets and stars as it was traveling through the solar system.  In addition, they had to calculate the motion of the earth, sun and other planets Voyager had to travel by.  This means the engineers basically worked with a three body problem or how to calculate a ships trajectory with reference to the sun, a planet, and an object which in this case is a space ship.

The data indicated if they sent the spaceship  near a planet, the planet lost some speed to the craft causing it to speed away from the sun without using additional fuel.  Back in 1965, the man who found the solution to the three body problem calculated the locations of Jupiter, Saturn, Uranus, and Neptune in the late 1970's.  He figured if the spacecraft was launched it 1977, using a slingshot path, it could avoid all four planets.  So Voyager was launched then and off it went, sending back information which provided new information.

On the other hand, the Kepler Space craft was sent out to observe stars outside this solar system.  Scientists are using the Titius - Bode equation to predict the distance of planets from the sun and is used to test the hypothesis that most stars have at least one to two earth like planets in their orbit.  In addition, This equation with a bit of an adjustment, has allowed scientists to find 228 planets unseen by the Kepler telescope.

If you want your students to experience finding unseen planets, check out this activity by NASA and JPL.  The activity uses pi and real data from the Kepler spacecraft.  The worksheet is more in the form of an infographic covering four different situations from find the radius of something based on the decrease in brightness, Jupiter and hydrogen rain, a seismic event on Mars, and Oumaumau (a recently discovered interstellar object.  In addition, they provide answers.

This is the fifth in a series of activities using pi and math in the program.  I'll report more on it tomorrow.  Let me know what you think, I'd love to hear.

Warm-up