Tuesday, July 17, 2018

Geometry and Mars

Mars, Red Planet, Planet, Starry Sky I recently started receiving updates from Space.com.  The other day, there was an article on how they used geometry to figure out something about the water on mars.

If you look at closeups of Mars, you'll see something that looks like scars created by running water. Scientists have argued about where the water came from and where it went.

A group of scientists decided to use a statistical approach to determining where the water came from. 

They looked at the angle that those scars crossed each other.  The angles indicate if the area is dry and where the water might have come from.

They discovered the angles are fairly low, indicating the channels were not formed by ground water. In addition, the narrow angles of the valleys indicate a desert climate such as one found in Arizona.  Wider angles of the valley would indicate groundwater coming up from the ground.

The best theory is the channels were formed by sporadic heavy rainfalls over a long period of time.  The rain falls and runs off quickly creating a network of these valleys. 

At this point in time, there does not appear to be any water on the surface but scientists believe the northern hemisphere contained an ocean about 4 billion years ago. It is thought Mars had an atmosphere which allowed the planet to have a water cycle. 

As water evaporated, it condensed around the volcanoes in the southern hemisphere before raining down and carving channels in the planet.  Unfortunately, it appears the atmosphere only lasted a few hundred of millions years before the atmosphere is lost and water disappears.

This is not the first time scientists have used geometry to determine things in relation to Mars. Back in 2013, NASA's Mars Reconnaissance Orbiter discovered the sand dunes in a crater are in a polygon shape.  In addition, these types of sand dunes are fairly common but only in the bottom of craters and other low lying terrain.

The micro climates along with heating and cooling causes wind to blow in certain directions to form these polygon shaped sand dunes.  Furthermore, the dark sand appears to be from an iron rich basalt such as that found in Hawaiian volcanoes.  We do not have anything like this on Earth.

If you do a search on this topic, you'll find several different papers on using geometry to help explore certain facets of Mars.  I find it quite fascinating.  I actually attended a talk a couple years ago in which a scientist compared geologic formations on Mars with the same type of thing on Earth. 

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

Monday, July 16, 2018

Kitchen Nightmares

Kitchen, Home, Interior, Modern, Room Over the past week, I have been doing a marathon viewing of "Kitchen Nightmares" with Gordon Ramsey.   The basic idea is he comes in to help a failing restaurant turn around.

Usually, the food sucks even though the owners state it is great.  There are always some type of personality conflicts between the owners, the owners and the staff, and sometimes between the owners and the customers.

Often the people who bought the restaurant have wanted to run one but do not have the experience or its been in the family forever and no one is willing to make changes, fire the chef, or step back.  The process is for Gordon Ramsey to come in and try out the food which is bad.  As he's commenting on the food, he gets more info on what's wrong with the place from the servers because the owners know its not them.

Then he watches the dinner service, checks out the refrigerators and freezers often discovering the food in them is in bad shape, or they've mixed the raw and cooked foods together, or it was all made days ago, or they use purchased food.  Of course the owners are shocked into changing by their workers comments and Ramsey gives the restaurant a makeover before they redo the menu and have the grand reopening.

Somewhere in the middle of the fourth season (there were 7), I wondered how many of these restaurants were unable to survive and had to close.  According to a Business Insider article from 2014, more than 60 percent of the restaurants closed while 30 percent of those closed within one year.  I wondered how accurate those figures were.

I found a site by the Kitchen Nightmare people which listed updates for every restaurant helped during the seven seasons of the show.  According to them, 57 out of 77 places closed, 18 are still open, two moved and two were sold.  There is information on each restaurant whether open or closed.  If it closed, the author provides a bit more detailed information such as one place closed because the state seized it for nonpayment of taxes.

Another site also has information on which restaurants are open or closed.  It contains a bit more information than the other site but the other one has it all in one place. 

The nice thing about both sites is that it provides enough information for students to analyze.  They can look at the 57 closed businesses and break that information down further into how long after appearing on the show they closed.  Students could also look to see how many of those closed down after they were sold, or did they close for some other reason than just went out of business due to being too far in debt.  As stated earlier one business was closed by the state while another closed due to serving liquor due to an expired license. 

The majority of episodes include information for the amount in debt each restaurant is.  Students could calculate an average amount for the restaurants although one business went into debt due to the wife remodeling the place.  Instead of the 350,000, she ended up spending 950,000 because she didn't bother asking the cost for each thing she wanted and her husband not saying no.

They could even look at each season and see which seasons had more businesses going out of business or staying in business.  It will change from season to season but does averaging the seasons together match the same for the seven seasons?  Pose that question to the students and see what they come up with.

Let students write up a short article outlining their finding from this exercise so they can learn to write reports to communicate with others.

Friday, July 13, 2018

Warm-up


The Math Behind Insurance.

House Insurance Protect Home Care Safe Han  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.

Thursday, July 12, 2018

Math and the Internet.

Monitor Binary Binary System Computer Bina  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.  


Wednesday, July 11, 2018

Ferris Wheel Math

Singapore, Ferris Wheel, Big Wheel  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.