Pattern Making

I have friends who work at FIDM which is the Fashion Institute in downtown Los Angeles, California.  This is where future fashion designers among others come to learn their trade. 

One thing they often comment on is that they've seen project runway or some other show and they want to be a fashion designer.

The issue is, they don't know how to sew, iron, or even cut a commercial pattern out. To be a fashion designer, especially if you are just starting out,  you need to be able to create the drawings before making the patterns.

To translate a drawing such as above into the pattern used to sew the final product, the process involves a certain amount of math. Creating patterns is sort of like looking at one of those complex shapes you have to find the area of, break it down into individual shapes before calculating the are for each shape.

 With patterns you have to take the drawing, break it down into pieces before figuring out the particular shapes needed to create the piece.  In other words, a student has to use critical thinking skills combined with problem solving to do this.

In addition, a pattern maker has to calculate the way the measurements are distributed in the pattern. For instance if the pattern needs to be made for a 36 inch bust, 27 inch waist, and 35 inch hips, the pattern maker needs to remember that the 36 inches does not divide equally into 18 and 18 because most women have a bit more in front so the back might be 15 inches across and 21 inches in the front.

Does the final outfit need to be fitted or is it loose.  If it's fitted, you might need to include fitted darts but do you need one from the bottom or one from the bottom and one from the side to make the bust fit better?  Most skirts if they are fitted have at least two darts in the front and two in the back.  Placement and depth of those are important.

What about the neckline?  If the neckline is a V-neck, the pattern maker needs to make sure the angle is correct, otherwise it won't look right.  If its a scoop neck line, the circular arc has to be proper or it won't hang right.  Practical applications of the use of angles and degrees.

After drawing the basic pattern, the pattern maker needs to make sure they include a seam allowance which is usually 1/2 to 5/8 in.  Commercial patterns have 5/8 usually while specialty designs are 1/2 inch. 

A pattern maker has to be good at math, including fractions because many measurements are like 25 3/4 or 32 1/2.  You might have to dive the fraction by 2 or 4 so you have to be comfortable using math.

Lots of math.  Tomorrow, I'll look at the math a fashion designer needs for a project from start to finish.  Let me know what you think, I'd love to hear.

Grid Patterns and Ratios

When I am not teaching math, I make costumes from scratch.  I also buy parts and put them together to create a certain look.  It all depends.  I was at Costume College this past week which is a three day event where people come to learn more about certain aspects of costuming.  They sometimes listen to lectures while other times, they attend workshops.

One lecture I attended was on enlarging pattern pieces drawn on graph paper in books.  Many times, the books are published showing a style like the one in today's picture.  On the other side of the page they print the pattern pieces. 

Sometimes, the pieces are separated while other times the pieces are printed on top of each other.  Either way, they are printed on a grid with a scale found somewhere on the page.  Occasionally, you will not be able to find a scale so you set one such as one square equals one foot.  Some of the earlier books do not even have a scale and you end up reproducing it on graph paper.

The instructor suggested people stock up on wrapping paper because the backside of it has one inch squares already printed there.  To do this, she taped a good sized sheet of paper on the board so it would remain stationary.  She would then place a dot on one intersection that represented the corner of the piece.

From there she would carefully count out from there the number of squares to the left, or right, up or down.  If there were curves, she'd carefully  mark off the appropriate dots and connect them using a french curve.  Eventually you end up with a larger pattern that has been proportionally enlarged properly.

Most people who start with these types of patterns, realize the enlarged pattern will probably not be the correct size even enlarged.  This is for several reasons.  One is that if the piece the author used was old enough to create the pattern, it could have been a young person's outfit so it will be too small.

Little history lesson here.  Way back in past times, especially in the 1800's once fashion and women's magazines started appearing, small patterns on grids were included in as a special extra.  Many magazines would publish one outfit a year, or every quarter or every issue depending.  Women were expected to take the small sized drawings and create the full sized patterns to make the outfit.

This is a wonderful example of how ratios and proportions are used in real life.

  Several of these dresses might have been made by enlarging a pattern from a book filled with patterns from a certain time period.  I've done it but I didn't use this method. 

Let me know what you think, I'd love to hear.  I'm still researching the mathematical pattern information for clothing patterns.  As soon as I find it all, I will be sharing.  Have a great day and sorry I went MIA for Sunday but I got so caught up in attending classes, I didn't get a chance to post anything.

Ratios and proportions

I am in Los Angeles learning more about historical, theatrical and other types of costuming.  Have learned a bit about math.  Will share more on Monday.  Already learned about scaling up a grid pattern.  Will share on Monday.

The Math of Eclipses.

  Tonight is the night of a lunar eclipse also known as a blood moon because the moon turns reddish.  The one tonight is due to be the longest one for this century and it's happening between two solar eclipses.

I wondered what type of math is associated with lunar eclipses  and I found several sites which can give students the opportunity to perform math associated with lunar eclipses.

As stated earlier, the next lunar eclipse is tonight but it cannot be seen in the United States.  Math and You have an exercise where the student can find out when the next visible lunar eclipse happens in their location.  There are also a few questions students need to answer based on the information they have.

This site explains how to calculate the length of a lunar eclipse.  It begins with providing some background information before providing explanations and the data necessary to calculate a variety of factors involved in this type of eclipse.  The nice thing, is the exercise breaks everything down and includes answers at the very end so students can check their work.

NASA even has a page providing the mathematical equations needed to calculate mathematically when lunar eclipses occur from 500 AD on to the future.  Many of these equations are polynomials and could easily be graphed so students can see how the graphs look and how to find the time the eclipse happens.

Since the site also has links to the number and length of each lunar eclipse, students could easily graph the points to see if they can find how often this event occurs.  From what I read, it appears most lunar eclipses occur in January and June or July but they do not usually last as long as the one tonight does.

This site explains what conditions are needed for a lunar eclipse, the view from space, and  the Saros Cycle which ancient people were aware of.  This is important because it shows students that just because the moon and sun get close to each other, they do not always cross paths and the way they cross determines whether you have a solar or lunar eclipse.

Sorry its so short but I took off again to Los Angles and will be home Monday night, quite late.  Unfortunately, I was awake all night, got to the conference so I could set up my area since I'm on the committee.  I am falling asleep as I write this.

Tomorrow and Sunday, I will have my usual warm-ups.  I'm hoping to come up with something really awesome from this conference.  Have a great day.  Let me know what you think, I'd love to hear.

Math and Music

I love math but I also enjoy playing music.  I usually play french horn in the local community band.  I've often wondered what type of connection music has to math.

It is well known that music helps teach fractions and ratios because its a part of learning to play music. That said, no one is sure whether musical training encourages mathematical ability or mathematical skill promotes musical ability or whether the skills develop at the same time.

Although there is some research to indicate that musicians score higher on standardized tests, it cannot be stated with total certainty it applies to every one. 

It has been suggested this occurs because most students learning to play instruments tend to come from a higher socioeconomic class.

Others have suggested people might have to have  high cognitive processing skills necessary to  drive both skills.  One example is the use of executive functions which helps people adjust to the changing demands of the tasks. Executive functions are defined as the ability to plan, organize, and complete tasks.

 Research has shown that having good executive functions is a good indicator of academic achievement.  Playing music requires a person to determine the key signature, tempo, repeats, etc before playing while completing key and tempo changes during the piece in order to produce the music. 

If you think about it, much of mathematical learning is looking for patterns, just like in music.  So many pieces repeat the melody throughout a piece or in the more daring ones, the tempo changes quite rapidly from 3/4 to 4/4 and back again. 

In addition, in the past music has been mathematically analyzed and created.  It has been broken down into waves each at a specific frequency.  Furthermore, people have noticed the relationship between numerical ratios and fixed sounds.  People have also applied group theory's framework of symmetry to analyzing music.

If you think of musical transposition, inversion, and retrogression as group operations, this can shed new light on older compositions.  It's been found that composers have used geometric type transformations as a device when creating music.  Even those who play the music often have to transpose or translate the notes because the parts are not written in the normal key.

Such a nice interweaving of math and music.  Let me know what you think, I'd love to hear. Have a great day.

Using Hand Gestures.

We all know that person who cannot speak without using gestures.  The gestures may be nothing more than hands moving in the air but the movement is there.

According to something I recently read, hand gestures in math help people learn.

One study done by Michigan State showed a video to 184  elementary students.  In the video, the instructor showed the property of equality such as 7 + 2 + 9 = 7 +   so both sides are equal.


The video was set up so the problem was shown next to the instructor.  In about half the videos, she used hand gestures as she explained it.  She used the left hand when discussing the left hand side of the equation and her right hand when explaining the right hand side of the equation.  In the other half, she just spoke.

When they tested the students, those who had watched the videos with gestures did better on solving the problems than those without hand gestures immediately after and performed even better 24 hours later. 

It is thought that the gestures help students understand the structure of the problem.  It helps them see that no matter what numbers are on each side, the results are the same.  The gesture of underlining the problem with your hand helps translate the concept from the numbers to the general idea which is what helps students learn better.

In addition, it is believed that students remember the gestures because when you watch someone else perform an action, it stimulates the areas in your brain that performs the action.  Something that helps create better understanding.

Another study indicates that if students learn to use gestures, it helps them learn the material better.  There have been previous studies which show people who spontaneously use gestures as they learn new material tend to remember it better in general. 

This was supported by a study of 3rd and 4th graders who were divided up into three groups.  The first group explained out loud their thinking and everything they were doing.  The second group explained and used gestures while the third group used gestures only without speaking.

All three groups performed equally well when asked to do math immediately after learning the material but the big difference showed up 4 weeks later when students were retested.  At this point the group who spoke and used gestures remembered 92% of what they had learned.  The group who used only gestures remembered 80% but the group who used only speech remembered 33%.

This particular idea is still being researched as they are trying to figure out why using gestures work so well in helping students learn the material better.  I'll keep following this one topic and see.

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

Knitting, Math, and Classes

  A friend sent me a link for an article dealing with a college math class whose instructor taught math in a very nontraditional way.  In fact, she threw out the textbook, pencils, paper, and calculators.

Much of the math is being taught through the act of knitting while recording thoughts and answers via blog entries.

One question she posed to students involved two different pillows. One had the sides in this color order: red, yellow, green, blue, while the other has colors in this order: yellow, green, blue, and red.  If you rotate the first one 90 degrees counterclockwise.  The question is "How many ways could you put the pillows down on the bed to make it look different.  The answer is 24 but only 8 can be found by moving one pillow.  To figure out the answer, students followed instructions to create the pillows needed to physically complete the exercise and find the answer. 

Another topic is Rubber Sheet Geometry in which the idea is that all polygons, if they are made of a flexible material, are actually circles.   This shows up in knitting because when you knit a sock or gloves using double pointed knitting needles, you use three needles to create triangular shape yet when you are finished you have something that is actually circular.

On the other hand, if they knit an infinity scarf, this shows them an exception to the concept of creating a polygon out of a flexible enough material and the result is a perfect circle.   They see it when they've finished following the directions.

In addition, there is something going on at the North Branch of the Nashville Library system to help girls use knitting to become more interested in math and science.  The group meets once a week on Saturdays to knit something in response to a challenge. 

The first thing the program does is to teach students to knit.  Once they've learned the basics students are given a series of challenges to figure out on their own.  One example of this is to knit a square made up of concentric squares of alternating colors.  Another one is for them to create their own bag which requires them to use proportion and ratios.

The youngsters wear cameras around their neck to record how they solve the challenge.  The visual record is then analyzed by the people running the program.

Another thing that has been discovered is that knitting often illustrates three dimensional mathematical concepts in a way that is easily understood.  One professor stated that he'd over heard two college students discussing analytic geometry only to find out they were discussing how to knit an argyle sock.

That is a cool way to teach mathematics but at the moment, I don't think it will take off the way it might because many schools seem to believe that math still needs to be taught in the traditional ways.  Let me know what you think, I'd love to hear.

The Cost of Reality TV.

  I discovered my Amazon Prime includes a few interesting reality shows.  The ones I  love are the cooking competitions.  Over the past few days, I've been watching tons of the shows. I am addicted to them.

In addition, many cable channels are moving from regular fiction shows to "Reality" or non-fiction television because people are attracted to these.

I've heard Reality shows are cheaper to produce than the usual scripted ones such as NCIS or Hawaii-5-0. 

Lets start with a few stats on reality television before we look at the cost.  In 2001, these types of shows accounted for 20% of prime time programming. It jumped to 40% in 2013 on average but for some stations reality television accounts for 90% of their programming.  In the 2012-2013 year, four of the top ten and eight of the top twenty-five top rated prime time shows were reality shows.

Due to the low production costs, the profit margins can be so much larger than with scripted shows. Several cable station's for profit margin (those with reality programming)
1. TLC with 60%
2. The Discovery Network with 58%
3. National Geographic with 52%
4. Investigation Discovery with 35%


The average cost for a reality show is between $100,000 and $500,000 which can be so much less than a scripted show.  To show the difference, Royal Pains, a scripted show, runs between $2 and $2.5 million per episode.  One reason reality shows cost less is because they use fewer writers and employees but the cost can be made up in paying the 'talent' hired for the show.  For instance, Sarah Palin wanted to be paid $1,000,000 per episode for her reality show.  That is the same amount James Gandolfini received his final year of the Sopranos show.

On the other hand, the girls in "Jersey Shore" a reality show, only received a few hundred per episode and eventually were paid $10,000 per episode as compared to Charlie Sheen in 2009 who earned over $850,000 per episode in Two and A Half Men.

Furthermore, these reality shows allow smaller or emerging cable stations can produce a full season for between one and three million because everything can be shot in one day.  If its a bigger network with more complex shows, which can cost quite a bit more.  For instance, one episode of Survivor can cost $2,000,000 per episode or a price of $30 million for the whole 15 episode season.  Each episode of Survivor takes 3 days to shoot with a pre and post production which runs most of the year.

Although I looked for math activities created to use information on the cost of reality television, I was unable to find any.  I wanted to share some of the differences between scripted and reality shows so you can share it with your students.  If you have ideas other than creating graphs or figuring out the difference in cost as a percentage, I'd love to know.

Much of the information come from this paper.  I found that one of the pie charts on page 12 had percentages that add up to more than 100% so there is a mistake.  This is the perfect opportunity to have them read the graphs and figure out what is wrong with it.

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

Warm-up

Warm-up

Scams and money.

As I mentioned before, I received two different texts on my phone offering me jobs which according to everything I read, are scams.  They are variations of the fake check scam where you are sent a check, you deposit it, withdraw a certain amount and wire it off to a third party, leaving you with a few hundred for doing very little.  By the time you find out the check is fake, the scammers have moved on.

I wondered how effective these scams are because I had a family member come to me about the text offering her $550 per week for driving her wrapped car around town.  I told her it was a scam so she wouldn't get ripped off. They appeal to those who want to make money quick.

I found a site from Australia called Scam Watch which provides data in chart form  showing the types of scams most common for 2018 up to the end of June.  They analyze the data or the year showing number of reports versus amount lost, the top 10 scams by amount lost or number of reports, amount lost versus method they were contacted, the age group of the victims, amount or number of reports by gender, and amount lost versus number of reports broken down by states.

You can find the information broken down by the year back to 2015,  by the month for each year,  by the type of scam or all scams or by both year, month, and scam.  Great data for students to take and enter into a spreadsheet to further analyze the data by comparing months, or years.  They could interpret the data to draw conclusions.

The above information but what about Americans?  According to Market Watch  the number of scams rose almost 60% over a one year period from 2015 to 2016.  In the same time period, it is said that one in ten have lost on average $430 to a scam to total $9.5 Billion.  In 2015, the average amount lost was $274 per person who participated in a scam.

According to another article from Market Watch, the amount of money lost to a scam increased 7 percent from 2016 to 2017.  In addition 40 percent who made fraud complaints lost money in the 20 to 29 age group as compared to only 18% in the 70 and older group.  Although fewer seniors lost money, they lost more at $621 versus the $400 for the younger group.  Over all, the number of complaints dropped from 2.98 million to 2.68 million, totaling $905 million, an increase of 7%. 

One last article is from the Federal Trade Commission's Consumer which lists information on the top frauds of 2017.  The number one scam was the imposter scam in which 350,000 people lost $328 million. This is where someone contacts you pretending to be a loved one, a government official, or someone else they are not. This is a great article to have students practice extracting specific pieces of information to interpret.

This is a cool, real life use of mathematics.  Let me know what you think, I'd love to hear.

Benford's Law and Securities Fraud.

  If you are a Numbers (The TV show) fan, you might remember this from Season 2.  Charlie used it to help track down a stolen DNA synthesizer in 'The Running Man'.

Your average person has no idea what the law states and what it is used to find but it is also known as the 'first digit law'.

Although this was first discovered in 1881 by an astronomer but it was rediscovered in 1938 by physicist Dr. Frank Benford.  Over a period of eight years, he analyzed a variety of data and discovered an interesting fact.

He discovered that the chance of a certain digit appearing in the first position depends on the number.  For instance, there is a 30% chance of being the first digit while there is a 17.5% chance that 2 appears first and so on.  The higher the number, the lower the probability it will appear first.


It's these probabilities that help investigators determine if a public company is more likely to end up in trouble with the Securities and Exchange Commission (SEC).  If a companies financials have different percentages than those from Benford's law, they are more likely to be caught for accounting irregularities.  It does not matter what the company produces, this seems to hold true.

This was supported when someone went through the companies who had been busted by the SEC.  The statistical deviation between the numbers submitted by those companies and Benford's law was 20 times the average for all firms.  That is significant.

When the companies who were caught restated their earnings, the numbers fell within the boundries for Benford's law.  It has been admitted they cannot catch every company due to the limited budget and employees by the SEC but it appears they catch the ones who are falling apart.

In addition to catching companies who are not using good financial methods, this law has been used to find voter irregularities, Greece's effort to hide debt, alteration of digital photographs and other forensic applications.

If you'd like to give your students a chance to use Bednord's Law, you could check out these three activities.  The first is from Cornell University.  It offers three different activities which utilize this law so they can see how it works in different situations.  The second is from Texas Instruments.  It has students calculate the percentages based on the rates provided. 

This topic came about as I looked for information on how much money is scammed from things like the car wrap and the mystery shopper scams both of which I received via a text.  In the process, I came across this and thought I would share it.

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

Mars Math By NASA

  NASA has created a whole series of Space Math books filled with a variety of problems for grades 3 to 12.  Today, I am looking at one of those books whose math deals specifically with the planet Mars.

Mars is one of those planets which has caught everybody's attention from the past to the present.  Even the cartoons had Marvin the Martian who gained quite a bit of popularity.

Although the ebook is not very thick, only about 78 pages, all activities are only one page long and include the answers.  The book includes a 'Math Topic Matrix'

If you look the lesson on 'Modeling the daily temperature of Mars', you'll see it covers averages, a graph or table analysis, coordinate plane/points, modeling, and trig.  Multiple skills to challenge the students and put a more realistic feel to activities.  It is not like the problems to practice the one skill found in a text book.

Another page contains the standards being met.  Although the phrasing is a bit older, it is easy to find the current Common Core or NCTM ones and use those. 

Each lesson includes the information a student needs to use to solve the problems given.  This lesson includes a line graph using the time in sols vs temperature.  The last questions asks students to predict the temperature at a time in the future based on the pattern.  There is a third page with the information on Viking 1 and Viking 2 so you could have students graph the information from Viking 2 before answering the same questions for this new graph.

One of the questions asks students to create a sine function to provide best fit for the information. As I said the answers are provided complete with the math calculations if you are not that strong in figuring out the equations. 

The activities are divided into three grade groups,  3-5, 6-8, and 9-12 but those are guide lines.  I've looked at some of the exercises in the first group and several of my students would struggle to complete them because they have issues applying fractions in this type of setting where they are given information as clues.  The clues are set up to tell you that something is a fractional size of another planet which is a fraction of a planet whose size you have.

I have to travel in August to a conference at the beginning  of school and I think I might just go ahead and use a few of these for math while I'm gone.  The sub can use the answers to help students complete the assignment if needed. 

Check it out and let me now what you think.  I'll be checking out other books in the near future and reporting back.  I'd love to hear back from you.

Geometry and Mars

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.

Kitchen Nightmares

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.

Warm-up

Warm-up

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

Warm-up

Math and Star Wars

  Star Wars, the first movie was released back in 1977 to huge audiences.  It made enough to lead to two more movies before a break.  Eventually the three movies covering the time before the originals were released.  Who can forget JarJar Binks with his bounciness and ability to drive adults crazy?  Now we are up to the 2nd of the last three in the 9 movie concept.  And - The series has spurned two other movies, Rogue One and Han Solo both giving more background to the Star Wars universe.

Fortunately there is a math activities available via Yummy Math to use to analyze the net profit based on production costs and world wide gross.  The activity also brings into account the idea of inflation and how the costs were different in 1977 vs now.  Back then a Hershey bar was only 20 cents rather than the $1.00 of today.  Students are introduced to inflation calculators so they understand the concept of in 2018 dollars when they see the comment.

We always want to incorporate reading into our math classes so students become better at interpreting what they read. There is this great article which does a statistical analysis of the different worlds found in the Star Wars movies.  They used Graph Theory to analysis the information to reach certain interesting conclusions.  For students such as mine who hate to read, you can create a sheet to accompany the reading so they look for specific information or the questions could require some mathematical calculations and conclusions based on the information.

Mathematic shed has a variety of activities for a variety of ages.  Some require solving equations while others require students to propose a hypothesis on what a certain number might represent.  There is also an infographic on the cost of the Death Star, students can use to find the answers to questions you create.  There are some for adding basic numbers for elementary students but I ignored those because they are a bit young for my students.

Then this site has a nice activity for creating a Star Wars Galaxy similar to the activity students do in the gym to set up the solar system.  The good thing for my students with this activity is that it requires students to convert from metric to standard which is something they struggle with.

This site has links to 15 activities based on Star Wars including a coordinate plane activity, fraction activity,  and drawing R2D2 which some of my students would enjoy.  Although many are geared for elementary students, a few are geared for older ones.

I may have to have a sub for the first couple days of class this year so I may set up a unit starting with one of the movies, followed by some of these math activities to ease my students back into work.

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

Futurama - Hidden Math Gems

Many of my students love watching certain cartoons such as The Simpsons or Futurama.  I know there is lots of wonderful math in The Simpsons but what about Futurama?  What math does it have shared in the 14 years it was on television?

There are lots of math jokes running through the show.  In fact almost every episode has a math reference, some are only known to mathematicians.  Three of the show's writers have PhD's including Ken Keeler who has one in applied mathematics.

Let's look at some of those jokes sprinkled through the series.  Some are puns and some are out right jokes.  So lets take a look at some of the wonderful references.

1.  "What's non-orientatable and lives in the ocean?"  A Mobius Dick of course referring to a four dimensional whale based on a Mobius Strip. 

2. In another episode, one of the characters sees 1010011010 written in blood on a mirror in a haunted castle.  Most people do not read binary but if you did, you would know the number is actually 666 associated with the beast.

3. One of the space ships has the registry number of BP-1729.  1729 is the smallest sum from two different positive cubes combined in two different ways.  The same number is assigned to a unit of soldiers.  Furthermore, it is assigned to the universe in Farnsworth Paradox. 

4.  A taxi cab has the number 87,539,319 which is the smallest number made from three positive cubes combined in three different ways.

5. In another episode, the artists snuck P vs NP on two books found in a bookcase of books.  P problems are easy to solve, NP are not but answers to NP problems can be checked.

6. How about a dating agency that advertises that is is both discreet and discrete.  Remember discreet refers to protecting client identities while discrete refers to data that does not vary smoothly or continuously.

7. Then there is the episode with the club "1^2   2^1    3^3 which is a play on Club 54, a very famous night club.

8. One of the writers ended up proving a theorem from the episode where the brains of two people were switched in "Prisoner of Brenda" but cannot be switched back.  The theorem known as Keeler's Theorem proved that no matter how mixed up people's brains are, all you need to do is bring in two new brains and everyone can get their own brain back.

9.  The local theater is Loews  ℵ0 plex with an infinite number of screens.  Alepha naught or ℵ0 refers to the different sizes of infinite sets which change based on the scale.  Since ℵ0 is the smallest cardinal number for the set, every one has an ℵ0.  

10. In an episode with a horse race, the track uses a quantum finish to determine the winner but all that means is when they measure results, the results are changed because its based on the observer. 

There are other references to Banach Tarski theoretical paradox, a beer called Klein's with an odd-shaped bottle based on Klein's bottle, and other unique things.  There are lots of fun references, many of which are only understood by math people straight off but knowing this, one can play clips from the show and then brig the math into the classroom this way.

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

Happy July 4th

Stats of Growing Old.

  The other day, in the local paper, I read an article on how old can people live too.  The oldest people end up living to 115 or 120 but the article had some awesome statistics on this topic. 

The article mentioned the Gompertz law formulated by Benjamin Gompertz in 1825 suggests the odds of dying double every eight years for people between the ages of 30 and 80.  After the age of 80, it appears that rates begin slowing down until they plateau between 105 to 110 years.

The law does not actually look at the odds of living to a certain age.  The odds of a woman living to the age of 110 is 2 in 100,000 but the odds of a man living to the same age is 2 in 1,000,000.  The odds are vastly different. 

Other odds given in the article state if you reach the age of 105, there is a 50 percent chance you will reach 106.  The 50 percent chance applies each year you live past 105.  So if you are 106, there is a 50 percent chance you'll reach 107 and a 50 percent chance you'll get to 108 years old.  Its the same as flipping a coin for heads or tails.

That sounds like a decent set of odds but once you add numbers to it, it isn't as good.  Think about it.  If only 2 men are alive at 110, that means 4 at 109, 8 at 108, 16 at 107, and continue to work backwards. It shows that not that many are alive at 105.

The article itself claims there are 3,373 women and 463 men in Italy who lived to the age of 105 and beyond.  These folks were born between 1896 and 1910.  There were four born in 1896 who lived to 105 or beyond while over 600 people born in 1910 lived to the same age.  That is a tremendous increase.  The article said the increase was due to  improved infant survival and the care of senior citizens so people could live longer.

There is an actual discussion on a possible limit of 115 years for a human life span but there are a few folks who live past that.  What ever the limit, if it exists, only a few can make it that long. 

The above is an example of statistics being used in the media.  Usually when I read something like this, I automatically begin researching the claims to see if the information is actually correct and explained in the proper context. We all know statistics can be misleading, so its good to check out the information.


We know what the stats are for Italy but what are the stats for the number of people who have reached 105 years old in the United States.  There are several people in my family who have made it to their late 80's.  I know of at least one person who made it to 97 before they died while another recently reached 94.  I don't know of many others who have made it that old.  I do know the odds are against his making it to 110.

For this type of article, I would provide copies of it to my students to read, write down questions they have on the material, think of what if's, before trying to apply the material to information they have from the census bureau.  They might create a spreadsheet to calculate the number of folks who live to 110 in several different countries.

It is important to read and think about articles found in the media because too many times, we accept the information without question and its not always presented properly. 

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

The Mathematics of Apps

  I've known for a while of one way to make money marketing apps but I recently learned of another way due to an article I read on Medium.

If you check out the internet, you'll find information on ways to make the app.  The ways include create the concept and write the app yourself.  If you can't code, create the concept and hire someone who will write the code for a flat fee.  Either way, you market it via iTunes or Google Play to make money.

Another way is to buy the original code for an earlier game that did well and has been withdrawn from the store.  When you buy the code, you agree to make changes so instead of a flying bird being the main object of the game, you make it a flying pizza, then market it.

The scenario of paying a developer and buying the code opens the door to calculating the break even point to determine at what point anyone starts making a profit.  It is not too hard to find the percent that iTunes or Google store take for marketing the app. 

In the scenario with the developer, there needs to be a bit of research done to determine the average cost of hiring a developer to write the code for a new game.  That is part of the start up costs.  For the other, the cost of purchasing the code is part of the start up cost.  In either case, the cost of getting the code done is the constant in the equation.  The next part is determining the percent the app receives from the app store is the coefficient of the variable in the linear equation.

In addition, you can have students create spreadsheets to change the selling price of the app to see what how the break even point changes for $0.99, $1.99, $2.99 costs.  Students could also create surveys of other students to determine which price most people are willing to pay.  They could also research the percent they get when selling the app at the iTunes store or Google play so they can use it in their spreadsheets.

A real life application of a basic linear equation for finding the break even point.  Every so often, there are articles of people who have created apps that succeeded so well, they were able to quit their day jobs and focus only on app development.  when students read that, they want to create their own apps to become rich but they have no idea how to determine break even points.  The above two scenarios help students learn more about that.

The other way to get money from apps is to create a service which requires the user to subscribe to the service to use the app.  The app is basically a portal to the service.  I read an article which analyzed one of the best selling apps and discovered the author threw together a bunch of things in one place but made the money on the weekly charge.  I think the weekly charge was $9.99 but over a year that is just under $520 per person. 

The article opened my eyes to reading all descriptions carefully before getting an app which indicates a purchase associated with it.  I've discovered many apps say the iTunes store will automatically charge so much per week against your credit card.  This is where people make the money is not through the app itself but through the weekly charge.

It is possible to create a spreadsheet to determine the amount made by looking at the different amounts charged for a weekly amount. This activity opens up the opportunity for students to conduct research to determine the general weekly amount and the types of apps that do this.

Real world applications at their fingertips.  Let me know what you think, I'd love to hear.

Warm-up