Warm-up

Early Female Innovator.

Hedy Lamarr

  If you said the words "Bombshell" people think of Jean Harlow, Marilyn Monroe, or anyone of a number of others from that time period but not everyone knows about Hedy Lamarr and her place in history both as a bombshell and inventor. 

She was considered one of the most beautiful women in the world.  She acted in several movies between the late 1930's and the early 1960's.

She originally came from Germany where she had a film career but after being discovered by Hollywood, she fled Germany for a career in Hollywood.

This is the side the world knew but there was another side that few people knew.  I only found out because of a television show I watched.  As soon as I heard, I checked the internet because I'd never heard of this and no one I knew had heard but she was also an inventer.

Back in 1942, she and composer George Antheil patented the design of a "secret communications system".  The idea behind this is that radio signals would change frequencies so the enemies could not detect messages.  They conceived the idea as a way to help America win the war.

 At the time the idea was patented, the system could not be built because the parts were not small enough.  The patent itself remained unused in the patent office until three years after it expired when the military used it during the Cuban Missile Crisis.  Their work formed the basis of the "spread spectrum" wireless communication used in blue tooth, secure wi-fi, and GPS.

Think of the mathematics involved in creating this type of system.  The mathematics and the understanding of engineering. She never really went to college so how did she manage this?  First of all, she was pretty good at mathematics but where did she gain her knowledge of the military and radio technologies?

In 1933, she married a Jewish arms manufacturer who converted to Catholicism so he could do business with the Nazi's.  She claims he kept her prisoner except for when he took her to business meetings and that is where she learned about military and radio technologies.  After four years, she left him, headed to Paris and obtained a divorce. 

Once she'd been signed to a contract in Hollywood, she met the composer and sometime during their conversations, the idea came up.  When they were granted the patent, they donated the design to the Navy to use but nothing happened until later. 

 The sad thing is she never received any monies for this.  It wasn't until 1997, she and George began gaining the recognition so rightly deserved.  I find it amazing that this woman, a Hollywood star, came up with such a far reaching idea that changed the face of the world.  She didn't have any formal training but still understood the concepts enough to do this. 

If we had to classify the mindset of this woman, we would classify her as having an open mindset because she didn't not cave to the idea that women couldn't do it.  I am amazed and thrilled to learn more about here.  Perhaps, I'll start a wall this year in my classroom showing people who succeeded in the world without formal training but still had the where with all to do it.

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

8 Ways to Make Lessons More Interesting.

  It is always a challenge to make lessons more interesting, especially math.  It seems that many times, we learn to teach the same way we were taught and the more modern books still follow the same basic pattern.

Lets look at some ways to improve your math lessons so students become more involved.  We have to catch their interest.  Remember, their minds will wander about 50% of the time.  These ideas are designed to help keep their attention for a longer period of time.

1.  Include a story so students have a better chance of relating.  The story might be about a basketball player (probability) or a famous chef, or even yourself.  The story can be about the origin of a concept, how someone grappled with it, or any thing to catch their interest.

2.  Create a hook. The hook could be a real world problem, a way of looking at a problem, or an interesting problem.  It could be a video, or other way of getting their attention.  "Teach Like a Pirate" is filled with ideas that can be adjusted to the math classroom.  I've made a few trailers to let students know what was coming in the next lesson.

3. Let students know the main points.  This should not be done once.  The material should appear throughout the lesson. Its important they know what is important in the lesson.

4.  Use images rather than words because the brain remembers pictures.  You need to connect the images to the story or the material. 

5.  Connect the math to when they will use it in real life so they don't have to ask the question "When will I ever use this?" 

6. Instead of going from abstract to concrete, begin with the concrete example so they can see how the math works before going to the abstract general formula.  Since students seem to be spending less time outside, they do not always have a grasp of certain topics like velocity, gear ratios, etc.  Its better to begin with the concrete examples.

7.  Use technology when possible for the drudge work in math such as finding statistical deviation, or the length of a curve. 

8. Ask more interesting questions rather than relying on the standard ones. As questions based on situations rather than just calculating  For instance, the questions could ask about the shape of a graph based on the acceleration of a skier going down hill.  It involves no actual math.  Simple sweet and easy.

We have to make sure we have our student's attention so these are some ways to make the class more interesting.  Let me know what you think, I'd love to hear.

Games and Retrieval Practice

  Retrieval practice is a wonderful way to help students shift their mathematical learning from short term to long term memory.  It is a way of recalling facts from your memory rather than looking at notes.

One of the best ways to do retrieval practice is to provide frequent quizzes so students have to practice retrieving the information.

One way is flashcards but most high school students laugh at the idea of those while if given a quiz, they may want to know how many points its worth.  Sometimes, it nice to sprinkle games in among the quizzes so students have a fun way.

One of the games my students love playing is Kahoot because they find it fun racing to be first with the correct answer.  It is a way to test their knowledge in a less stress situation.  Even the students who struggle, enjoy playing the game.  You can hear when they missed a question because it provides immediate feedback.

In addition, I can tell from the leader board when no one gets the correct answer.  This tells me, its time to work the problem to remind students of the process.  Unfortunately, the bandwidth at school often prevents me from playing the game on line.

For days like that, I pass out white boards or have students grab an ipad, pull up a drawing app so they are ready.  I write a problem on the board and watch them go.  When they are done, they hold up the work, I check it and respond with a thumbs up or down.  Thumbs up means its right, thumbs down , its wrong.  They want to be the winner because I offer candy for the right answer.  They will do anything for candy.

Another game is a variation of bingo but let the students choose the answers.  You provide a list of answers from say 50 questions for a 5 by 5 bingo card.  In the bucket, you have 50 different problems.  Select one problem at a time, give students a chance to work the problem and find the answer.   Once a student has 5 in a row, they've won.  The 50 problems should contain a nice mixture of easy, medium, and hard problems to everyone is challenged and those who struggle have success.

Yes, you have to work the problems out ahead of time to know what the answers are but once you have a choice of 100 problems, you can mix and match as needed.  The problems can cover several different topics so students have learned. 

Just make sure the games chosen allow students to work individually or in pairs.  I tried having students play Jeopardy in groups but it didn't work well.  If they were in groups larger than pairs, one person often did the work while others didn't.  I also give points to those who all had the correct answer.

I try to play games at least once a week so students get the retrieval practice they need.  Why not, if they are having fun and learning at the same time. 

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

Heidelberg Trolly

Yesterday, I shared some real life problems based on three wine barrels at the castle including the largest one in the world.  There is a cable car to take from the base up several levels to the absolute top.

The reason I'm looking at the cable car is because the slopes for this are way different than those found associated with American roads.  The maximum for American roads is 6 percent or up six feet for every 100 feet. 

The trolley or cable car makes a stop at the castle level, at the general view where you change trolleys or the top.  Based on what I could see, it looks like the slope for the first two parts is about 22 %.  The last part, the wooden cable car had an even steeper slope.  It began at 22%, moved to 28% and ended up at a 40%.  Going down the hill, looking back up, I could actually see the curve in the hill as the slope increased.

This opens up some interesting types of open possibilities for the math class.  Think about telling the students there is a 22% slope but what possible numbers could the rise and run be which produce the 22% slope or what possible numbers would produce the 40%.
 
Real world math right here.  Once students have come up with possible choices for the rise and run, pull out topographic maps for students to find the actual slope for each part of the trolley.  Once students have these numbers, students can look at other trolleys , other cable cars both aerial and land based to see what type of slopes they have.

Final part of this is to choose a spot which requires a trolley or ski jump to get from one point to another.  The location should have several possibilities for it so no one choice is right. Break students into several groups before providing the students with pictures, topographic map and ask them to provide a design for that location.   They need to create a full presentation from a drawing of a the trolley or ski jump or cable car, to reasons why it was done that way.  They are trying to sell the investors on selecting their group and their design for the project.

I love traveling because I see all sorts of interesting mathematical problems wherever I go.  Let me know what you think, I'd love to hear.  Have a great door.

Wine, Wine Barrels, and Cylinders

While visiting Germany, I visited a castle in the middle of Heidelberg.  The castle is known for several things including its Apothecary Museum but that is not for today.  In another part of the castle, with the restaurants and wine bar, there were three wine caskets of varying size.  When I looked at them, I realized this is one of those perfect opportunities to bring back a real world use of cylinder. 

On display, just past the entrance, stood two wine barrels, a small one and what I thought was the big one.

I was only able to find information on the size of the largest barrel but I'm going to assign values to the others so I can create problems using the two other casks. Based on research, the smallest is probably 23 by 28 inches or about 65 gallons.

The middle sized which is the one off to the right is probably 4 meters across by 5 meters long so its easy to calculate volume on both.  Its easiest to make convert all measurements to centimeters because 1 milliliter (ml) = 1 cubic centimeter (cc)= 0.001 liters (l) = 0.000001 cubic meters (m³).

Add in these conversion factors and students can calculate how many bottles of wine, each one can fill. 

1 ml = 0.061024 cubic inches (in³) ; 1 in³ = 16.4 ml

1 ml = 0.000035 cubic feet (ft³); 1 ft³ = 28,317 ml

1 ml = 2.64 x 10^-4 U.S. gallons (gal); 1 gal = 4.55 x 10³ ml

This turns out to be the largest of three.  It has a diameter of 7  meters and a depth of 8.5 meters.  This is enough information to calculate the volume of the worlds largest wine barrel.

In addition, it holds 220,000 liters or 58,124 gallons of wine.  This information allows students to calculate how many 750 ml (25.4 ounces) bottles can be filled from this cask.

At least three problems from these pictures of wine casks found in a castle in Heidelberg which require calculating volume, converting, and figuring out how many bottles can be filled.

All real life problems.

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

Warm-up

Warm-up

Different ways to Quiz

  As we all know, providing chances to review the material in the form of quizzing is important but we also know students do not always want to do the same thing every times, especially if they are being quizzed at least once a week.

I'm looking at different ways of quizzing students so they have the opportunity to try a few different ways and perhaps not get bored.

One method is where students solve the problems on a short quiz individually.  Then match students up in pairs so they can check each other's work.  Next, put them in groups of four to check the work, then into groups of eight and continue till you have one huge classroom sized group.  Once everyone has had a chance to check each other's work, students turn in the quiz.  The whole class receives the lowest grade.  This method encourages collaboration, self-checking, and students learn to be a bit more careful with their work.

The next choice is to create groups of four students.  Give the group a short four question quiz for them to work out all the problems.  The next day, take a 4 question quiz cut into strips so each person in the group has a question to complete.  They turn in the completed questions and the group gets the grade from the combined problems.

Another possibility is to provide individual students with the quiz to complete on their own.  Then students divide into groups of two.  Each student gets a new, clean copy of the quiz they work on together in their groups of two.  Once completed, they turn the pairs quiz in for the grade.  As a teacher, you can choose to grade both the individual tests and the pairs tests separately or weighted so one is worth more than the other.

A variation on the above ideas is to create two separate short quizzes passed out two partners.  Each student takes their own quiz.  When done, they compare answers so if the answers are the same, they know their problems are correct.  If the answers are not the same, they know they need to check each other's work to see if they can find the mistakes.

Frequent quizzing provides students with retrieval practice which helps them move the information from short term to long term learning.  In addition, it allows them a chance to become more proficient.

Sorry its so short but I just flew back from Germany and I am exhausted.  I did not get a chance to get much done while visiting relatives and attending a wedding.  I should be back to normal on Monday. 

Let me know what you think about these ideas on providing quizzes to students.  I'd love to hear.  Have a great weekend.

Volcanic eruptions and Flow

  

I’ve  been reading articles on the volcanic eruption in Hawaii because I have land in the area.  Right now, it is not in any danger because most of the lava is escaping further south along various fissures.   At one point in the article, a scientist stated lava was moving at a speed of 17 mph.  This is a starting point for creating some real world problems involving a real world event.  Imagine, if this math could be combined with a unit taught by the science department!

There are 4 things which effect the speed of lava.
1. The composition of the lava and the amount of gas.  In other words it’s viscosity.

2. The amount of material coming out do the volcano.

3. The slope of the land.  The steeper the slope, the faster it will flow.



4. Whether it’s flowing in a sheet, a channel, or a tube

If lava is traveling fast, it moves about 17 mph while a more realistic speed is 6 mph.  Lava can get down to half a mile an hour if it’s quite viscous.  Now consider the average speeds of a human.  They can run at 20 mph, jog at 7 mph and walk at 3 mph.  This means it is possible to outrun any lava flow.  There are a few flows that get going extremely fast but I gave average speeds.

Now if students check out a topographic map to check out the slope of the area with current volcanic activity in Hawaii, they can get a better idea of slope, shape of the land the lava travels across to get a better feel for why the lava is going 17 mph.  I just down loaded an app which gets me access to topographic maps.

The teaching engineering site has two lovely lessons which contribute to a student,s understanding of how far lava flows and how to find the speed of flowing lava.   The first exercise has students learn about fluid flow in regard to volume, slope, viscosity, and surface through hands on experimentation while the second continues with how volume, viscosity, and slope effects the area covered by lava.  Both of these exercises require mathematical calculations.

In addition, one of the places in Australia offers a lesson on the speed of lava.  I think it is good to offer some lessons as activities or experiments sine it adds a real world element to the whole think.  Besides, someone will ask how they managed to calculate the speed of lava if it is so hot.

This provides the basis for a wonderful cross curricular activity among English, Math, Science, and Social Studies because they can look at so many facets of this one topic.

I will be back in the states by the weekend so I will be back to using my computer instead of using my iPad.  Let me know what you think.

Taxes

In the United States, most places have a sales tax charged on when anything is purchased.  In some places, sales tax is not charged on food items.  Of course, we have lessons designed to remind students of how it is calculated.

In Canada, there is something similar to a sales tax but it is referred to as a ‘Value Added Tax’.  The last time I went to Canada, the VAT was added when I checked out.  Then when I crossed the boarder, I stopped at the duty free place and filled out a bit of paperwork and got all the VAT back.

Now that I am in Germany, I’ve learned they also charge a VAT but it is included in the price shown rather than being calculated separately.  That makes it easier to calculate if you have enough money. According to my niece, the VAT accounts for 19% of the price so if I purchase something for a Euro, 19 cents is the tax.

Now think about how often we have students find the amount of tax due on purchases.  We do not give students enough practice in determining the tax rate nor do we provide a total amount and a tax rate so students can calculate the amount spent on tax.

Most students would claim 19 % is high but when we discuss taxes, we usually look only at sales tax. We do not look at road taxes which can be quite high or the taxes on cigarettes, especially in New York City. Do we ever take a moment to enlighten students on property taxes, inheritance taxes, windfall taxes, or any number of taxes most of us pay as adults?

Many of the above mentioned taxes are included in the price of things already but we do not realize it.  One such example is gas.  If we want students to understand more about taxes such as these, show them how some states have added taxes to interesting things.

A few years ago, I was in San Francisco when taxes were added to fines so most tripled. An example would be the $17 jay walking fine that suddenly jumped to $51 when they added a %200 percent tax to it.  The last time. I stayed at a hotel, I found a strange charge on my bill. It took the front desk about 10 minutes to get an explanation for it. Apparently, the city of San Diego had levied a tax on the total taxes charged.  In other words, they taxed the taxes.

I have a friend who is building a new house near San Jose. He commented he is being taxed for all the wood he purchases for his house in addition to paying a sales tax on it.  I realize we want students to be able to calculate tax and tips but it wouldn’t hurt to expose students to other taxes so they have a chance to transfer their knowledge.

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

Metric in the United States.

I do not remember if I mentioned it but I am visiting Germany for my nieces wedding.  It is hard to miss all the electrical lines crisscrossing the country side.  Many of the trains are powered by electricityl lines overhead, in much the same way they did a long time ago.

I realized we do use metric in the United States but we often times don’t give it much thought. If you ask most people to give examples of how metric is used here, you would get funny looks from people because they have not connected metric with this country.  So let’s look at some places we see it here.




1. Most power bills shows the amount of electricity used is in KWH or kilowatt hours.  Kilowatts or 100. Kilo is usually associated with grams, liters, or centimeters. Every bill will have this somewhere.

2.  When you buy soda pop, you can buy the drink in one or two liter sized bottles.  No one thinks that is strange.

3.  Most prepared foods have their weight in both American and metric measurements. If you do not believe me, check your cans of soda.

4.  Most cars sold in the United States have speed listed in both mph and mph. I’ve had to use the mph when I’ve driven through Canada.

5. Most people including myself have a set of metric tools somewhere around the house for when you work on your foreign car.

6. Many track and field events along with swimming and other sporting events are recorded in metric because competitors must be using the same unit to compare trials accurately.

So when we teach our unit on metric, we need to create an activity which would have students researching how and where metric is used in this country. In other words, they need to learn they are not totally isosolated and do use it.

Maybe this way if and when the United States joins the rest of the world in being metric, the conversion will not be that hard because students will already be halfway there.  Let me know what you think, I would love to hear.

Different Mindset

Mindset is something important when it comes to learning in general. We talk about students having a closed mindset if they think they are unable to learn math since they are not good at it or do not have the natural ability.  It has been found with an open mind, they are more likely to do better in math.

Unfortunately, there is another area in which the open or closed mindset is found within education. It is found in some students, teachers, and administrators.  This attitude has to do with the value of incorporating technology into the classroom,

Although we know technology is neither good, nor bad, it is how it is used within the classroom that counts.  We cannot just throw technology into the classroom because it is there. We have to have a purpose foe every use of technology when integrating it into our lessons. 

In addition, we have to prepare students for jobs in the future without knowing the exact skills required except, those jobs will require technology, collaboration, and communication.  It may require collaboration among coworkers who live in different geographic locations.  It might require the use of virtual rooms.

We need to make sure students have the knowledge and ability but if many of the students, teachers or administrators have a closed mindset, it makes it harder to imbue students with the knowledge to compete in this new world.

 Many of my students have developed the mindset that their learning can only be accomplished if they are using a computer. If they do the same assignment on a device such as a tablet, they argue they cannot learn the material not realizing that both are powerful computing devices.

I have been told by administrators I use to much technology in my classroom.  I should not have them watch annotated videos, do collaborative assignments in google slides.  I have been told my students learn best from the books and I should be hitting the books hard.  I have also been told that by allowing students to use technology, I am giving them the opportunity to get all the answers off the internet.

I have had other teachers tell me that students learn best from reading and doing work from the book.  No other way works as well.  They have also insisted that I am harming them by requiring students to watch annotated videos or ask them to write an explanation for why they chose to solve a problem the way they did.

To me, all three situations show a closed mindset because they are restricting their learning.  They are stunting their growth at a time when they need to expand their repertoire of skills for a rapidly exploding world of technology,

It reminds me of a couple of lines from a song by Bob Dylan. ‘The old roads are rapidly changing, get out of the new one if you can’t lend a hand for the times they are a changing’

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

Flew over Greenland

In transit

 
Just arrived in Germany. More tomorrow

Math and The Law

  When I was researching more on the math used by forensic scientists, I came across a couple things on math and the law which I found so interesting, I thought I'd share them with you.

Let's start with the Amanda Knox case.  In case you don't remember, she and another person were prosecuted in Italy for the murder of another person.

After being convicted, they appealed the court decision.  They won that but then the acquittal was thrown out over mathematics.

Their conviction was based on the identified DNA on the "murder" weapon.  The sample was quite small and back in 2009 the testing at that point could not provide a reliable result.  In 2011, when the appeal hit the courts, the judge decided that retesting the material would not provide any better results because the sample was smaller.  He ignored the part stating, testing methods had improved to the point of increased reliability of the results.

The court overturned the overturn because they felt the judge did not understand that if a test is run twice and it yields the same results, we can assume the first test was reliable. 

In another case, a Dutch nurse was convicted of murdering people on her watch.  The prosecution witness used incorrect methodology to state there was a 1 in 342 million chance of the deaths being natural.  So the jury convicted her.  After a long fight, that included a panel of statisticians, it came out the odds were wrong and the deaths were natural.

Another case of wrongful conviction based on bad math, involved a young English mother convicted of murdering her two children, both of whom died of crib death.  The conviction was based on a prosecution witness who claimed the chances of this happening in one family was one in 73 million.

Again, after a long fight and the involvement of the Royal Statistical Society, it was shown the basic method used to provide the probability had been flawed.  The women's conviction was overturned but the experience changed her.

Another case involved a couple who were convicted of a robbery based on the incorrect assumptions  applied to the multiplication  rule of probability as explained by a mathematician. The basic probabilities were calculated as independent events rather than conditional.  In other words, they treated The female who was blond and wearing a pony tail as 1. a white female with a pony tail and 2. A white female with blond hair rather than a white female who wore her hair in a pony tail. They were able to show the flaws in the math, and the conviction was overturned.

These are some ways math has been used incorrectly in trials.  Its a nice way to show how math can be incorrectly used to convict people.  Let me know what you think, I'd love to hear. 

Hawaii, Volcanoes, and Loss

  I have spent the past 11 days or so in Hawaii for a conference and for holidays.  If you've been listening to the news you know that there has been lava flowing down to the sea on the Big Island.  Its wiped out numerous houses and closed multiple roads.

The reality is that only a small part of the island is actually effected directly but the air quality may be what effects most people who are not directly harmed by the eruption itself.

While listening to the news the other day, they indicated that Hawaiian Airlines has cut the number of flights from Honolulu to Kona and Honolulu to Hilo back each by one flight per day.  The news also reported that many of the Papaya farms have been destroyed so there is an expected shortage.  Furthermore, since so many houses were burned or covered, the government will have fewer properties available to collect property tax from.

So what are other ways this area is suffering losses.  Several of the cruise lines are cancelling stops in Hilo, staying an extra day in Maui. Many of those who come ashore, take a trip up to the Volcanoes National Park but since the eruptions began, about two-thirds of the park has been closed.

One figure given is the park closure has cost the Big Island almost $166 million in revenue.  The same source states the amount rises to $222 million if you include the 2000 jobs indirectly effected by the park closure.  Last year, the park attracted 2 million visitors out of the 6 million who visited the island.

In addition, the number of bookings for tours and lodging has dropped 50% since May.  This can be devastating to residents because approximately 30 percent of private sector jobs are in tourism.  That is just under one-third of the jobs.  Even places in Kona are experiencing a reduction of bookings.

Furthermore, all guest houses in the area have had to close down, causing them to loose income and visitors had to find additional lodging away from the area.  Unfortunately, these cancellations and closing also effect businesses to the point that many are having to layoff staff or even closing.  Its a ripple effect.

When we hear about disasters such as this or the huge tidal waves that hit Indonesia a few year ago, it opens up the opportunity to have students look at the real life economic losses associated with these natural disasters.  This is a wonderful example of real life math so students can see real applications of what they've learned.

Its a perfect project which combines research, math, writing, and connecting it all together.  These are the types of things students need to do rather than only look at the math taught in a classroom away from real life.

Let me know what you think.  I'd love to hear.  If you have ideas on this type of project, please share.  Have a great day.

More Math in Forensics.

  Since I've spent quite a bit of time watching Forensic Files during my visit to Hawaii, I wondered what other math they use in Forensics beside blood splatter and height based on shadows.

In general terms, crime scene technicians must use precise measurements so they can eliminate those.  For instance, if they know the length of the shoe print, they know the size of the perpetrator based on the length of the shoe.  I recently ordered a pair of shoes from a company who requested I measure my foot in centimeters so I could order the correct size.

In addition, proportions are frequently used in forensics to provide additional information.  An example would be if a human leg bone were found, they could then use mathematical equations to figure out what proportion of a person's height is before calculating the actual height. They can also determine if the person was a child or adult.  Furthermore, they can use the length of a person's stride to help determine their height.

Forensic scientists often state their results in probabilities such as the DNA sample matched the DNA found at the scene so the chance of the person doing it was 1 in 2 billion people.  This provides juries with a better idea of how accurate the match is.

Specific examples using mathematics in forensic scientists vary.  One way is to use the ratio of the diameter of the medulla or center pigmented part of the hair to the diameter of the whole hair strand. 
If the ratio is less than .5, the hair is human.  If its greater than .5, the hair is animal. In addition, by careful examination, it is possible to determine the type of animal the hair is from.

To determine if the pelvic region of a skeleton comes from a male or female, examiners check the angle between the inferior public rami or the angle in the front between the two halves.  If the angle is an acute angle of about 70 degrees, but women have an obtuse angle of 90 to 100 degrees.  Forensic scientists often use a protractor to determine the angle.

When comparing fingerprints, they look at distances between grooves, and certain obvious patterns.  Distance indicates measurements because the anomalies must be exactly the same distance on both prints or they may not be considered a match.

Lie detector interpretation relies on blood pressure, pulse rate, and breathing while the length of a skid mark plugged into a formula can determine the speed of the vehicle.  This one I know about.  A family member was sited for speeding after hitting a street sweeper that was performing a U-turn just past a blind curve.  We went and measured the length of the skid marks, I looked up the formula so I could check their calculations. They were wrong in their calculations.  I proved he was not going more than 37.5mph in a 35 mph zone. They came back and charged him with negligent driving but at the trial,  the ticket was thrown out due to being totally prepared mathematically and scientifically.

There is so much more they can determine based on the evidence and through the use of mathematics.  This is just a bit of the math used by forensic scientists.

Population Growth.

  The other day as I sat in the lobby of the hotel, doing some people and bird watching, I wondered if I could present population growth in a slightly different way than normal.

Since I work in a small village with a population of between 900 and 1000 people population growth is not as evident. If school numbers are indicative of the population, it has been following a fairly circular increase and decrease, never going over a certain number.

I'd love to find enough information to have students calculate the population growth of the village but it has stayed pretty steady with the number of people who leave the village and those born. I'm not even sure where I'd find those numbers.

I know it is traditional to use bacteria or humans but I think for my students, if we did popular growth based on animals such as village dogs, moose, or other local animal, students might relate to these a bit more before I try to introduce human population growth.  In addition, it would be good to discuss what happened to that deer population when they removed the predators and the results.

Too often we teach population growth as if its nothing more than a mathematical equation or set of equations.  It's important to show students that as population increases,  there is an increased consumption of materials, production of trash, and an increased production of food with a decrease in available fertile land.

When we apply the formula to the growth of bacteria, we need to explain why it is important to know how to calculate this.  Ask students if an increase in bacterial population coincides with an increase in disease.  Do outbreaks follow the tendency for populations to outstrip the production of food and resources before declining?  Does consumption increase at the same pace as the growth of the population or is it faster?  

I'd love for students to understand that although the population is increasing right now, it is not increasing at the same rate equally around the world.  It would be nice to break down the daily increase to see where most of the growth occurs before having them venture guesses on why it happens that way.

This is one case where it is easy to show connections between population, consumption, food production, and how patterns of growth change. 

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

The Cost of Parades.

Pan Pacific Parade

Living in the bush of Alaska, we really don't have parades but most of my students have watched parades on television.  I know they've seen Macy's Thanksgiving Day parade and the Rose Bowl Parade.

Hawaii is noted for loving to give parades and there were two different parades this past weekend.  One was Saturday while the other was on Sunday.

While watching the parade, I realized there is a cost involved in putting one on.  I know I've never thought about the cost involved in a parade.

The cost of a parade will vary according to what is in one.  If you look at Macy's Thanksgiving Day Parade, the cost is between 10 and 12 million dollars each year.  Some of the costs of this parade are as follows:
$190,000 for each new character balloon but the cost is only about $90,000 if the balloon is reused.
$510,000 or more to purchase the helium for each balloon.
$30,000 to $100,000 for the construction of each regular float.
$780,000 to $2.6 million for each of the fancy artistic floats
$2,000,000 for the costumes including the Santa Claus costume.
$139,000 in property taxes for the Parade studio.
$1.3 million for the employees who work with the parade directly.  This includes 26 full time and 14 part time staff.
$0 for the police force as the city provides their service for free and the parade is run by over 8000 volunteers.
I don'r know how much it costs for the drivers for all the floats, medics in case of emergency although they might be volunteers,  and other incidentals.

Of all the research I did, I could only find costs for Macy's Thanksgiving Parade but many places host at least one parade a year.  I know Fairbanks, Alaska has at least one parade in July where they shut down the roads between the Carlson Center and goes winding through downtown to the Coop.  I don't know what it costs but I suspect it would be easy enough to find out by calling the city to ask. 

For the Fairbanks parade, most groups decorate their own float which is pulled either by a professional truck driver or a member of the group who uses a personal pick-up truck.  Many groups walk with signs while others drive antique cars.  I believe most entries pay a small fee to enter the parade.  I know the cost is no where near what it costs in New York City.

This could actually make a nice project for students.  Have them estimate the cost of running a parade in your town including permits, police, floats, etc.

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

Warm-up

Warm-up

3 Dimensional Snowflakes.

One of the sessions I attended at the Educational Conference was on 3 dimensional printing. It was awesome because I learned a bit about the math involved in creating the snowflake to the left.

I created it in a free online program called Blockscad.  Fortunately, the person leading the session walked us through the process step by step so we'd get  a proper result. 

I will tell you I forgot to pick up my finished product, otherwise I would show it to you via photograph.

To the right, you see the code we wrote for the snowflake.  Basically, I had to give the number of shapes which in this case is a square, I needed to produce the final product.  Then I had to find the angle rotation for each square by taking 360 divided by the number of shapes.  So this would be 360/10 or 36 degrees.

That is what the two brown ones did.  The purple tells the computer to go from 0 to 360 using a jump of 36 degrees.  You have to tell the computer how much to rotate the figures and move them. 

Immediately, you have a real life use of transformations in a situation students might actually relate to especially if they know they can print the final product once its done.  The instructor had us create a square by finding the difference between two cubes so only the x and y coordinates actually had the difference while the z coordinate has to be zeroed out.  As far as I can tell, the difference in the x and y axis gives the thickness of the line. 

It was fascinating to play around with the rotation and translations because one controls the distance between the "floor" and the top of the snow flake while the other sort of changed the flatness and created more depth.

If I had not had something to follow, it would have taken me a while to actually figure everything out.  As I worked this, I got to thinking how much fun it might be to do this as an art project where they students start with the square and have to create their own unique snowflake.  Imagine having them determine how far to move the square each time. 

I'm off to write a lesson for my geometry class based on this type of activity.  I love attending conferences because I end up with lots of new knowledge and ideas for my classes.  Let me know what you think, I'd love to hear. 

Have a great weekend. I'll be spending it watching hula, hula, and more hula.  Let me know what you think. I'd love to hear.

Parabolic art, coordinate system and spirals.

At the end of the Kamehameha Technology conference a few of us discussed my presentation on incorporating art into the math classroom.

One of the people realized the foundation of the piece of parabolic art to the left is the x-y coordinate system. 

It would be so easy to write directions up using coordinates to describe the placement of the lines running from the x axis to the y axis.

Each line would have two coordinates associated with it, a beginning and an end.  In addition, all the coordinates would contain a zero for one of the coordinates since the lines begin and end on the axis.  Admittedly, students would need to use graph paper for it, either actual paper or digital paper.

This gives students the opportunity to practice their graphing skills, remembering how to read coordinates, and produce a piece of art.  If you would rather not use the x and y axis as the starting point, you could just as easily translate or move it 5 units left and 3 units down so the center is at (-5,-3) instead of (0.0).  This way you are introducing transformation into the design and practice.

In addition, it wouldn't be that hard to include the shrink and stretch so one axis is twice the length of the other.  This would mean you might use 1/2 inch distance on the stretched axis and 1/4 inch on the shrunk axis.  You could also have students rotate the figure to add another element of transformations to the activity.

So one activity can be used to have students practice using the coordinate plane and play with transformations to create art work.  These ideas came out of a short discussion among three math geeks who were thinking of ways to make the art as part of the actual lesson.

Another topic we discussed was starting incorporating a Pythagorean Spiral in a geometry class by having students use compasses to draw the perpendicular line to create the 90 degree angle. 

In addition, the leg of the next triangle can be found using a compass or protractor depending on the skill required.   Students can also calculate the hypotenuse for each triangle in the spiral. 

It would be possible to include a short lesson on the Nautalis and the mathematics involved in it.  It shows a connection between real life and mathematics which is something students really need.

Again the above ideas came after my presentation from other math teachers who are taking the basics and adjusting them to meet the needs of their class. 

Math teachers rock.

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

Drones and Math

  I am in Hawaii to present at and attend the Kamehameha Schools Educational Technology conference.  One of the talks I attended was an introduction on drones.  During her introductory presentation, she discussed a few mathematical concepts and I wondered what other mathematics could be used with drones.

First she mentioned they had to use trig to figure out how to have the drone fly to the exact same place every hour so photographs could be snapped. 

The idea was to see the progression of smog as it settled down around the city.  Once the photos were taken, they were put together to create a longer piece, like a video so people had a better idea of smog movement.

In the class, she had students create a flight path mimicking a regular pentagon.  That involved a lot of discussion on the best way to accomplish it.  One student finished much earlier than the others so she had him create a circular path because she thought it was not possible.  He made a 360 sided polygon and used that for the circular path. 

I can't remember which mathematician used the idea that if you had a polygon of enough sides, you'd end up with a circle.  I know that many Alaska Natives use this technique to create patterns for circles.  To me this was a valid way to attack the problem and his final product did look like a circle.

Just these three topics came up in her short 50 minute presentation.  It seems to me that there is a lot more mathematics involved such as:

1. Addition, Subtraction, Multiplication, Division, and Circles.

2. The Pythagorean Theorem and Trigonometry.

 

3. Vectors.

 

4. Linear Algebra and Coordinate Transformations.

 

5. Differential Equations.

 

6. Fluid Flow and Airfoils.

 

7. Statistics.

 

So many different types of mathematics found in using drones.  This  is another way to make mathematics more real so they want to learn.  

 

Let me know what you think, I'll share more things I learn tomorrow and I'll return to this topic a bit later on.  Have a great day.

Pythagoras Caught The Murderer!

  Yesterday, I was finishing up something while watching something called Forensic Files on TV.  It caught my attention because they were trying to find enough evidence to convict a couple of people of murder. 

There was a photo of the man throwing the murdered victim off the cliff but the woman involved claimed she'd never been anywhere the murder.

The forensic scientists went back over the evidence and in one photo, they found a shadow of the photographer.  The guy stated they used Pythagorean Theorem to figure out the height of the photographer.  

Digital photography time stamps pictures so the police knew the date and time of the photograph. This gave them enough information to determine the location of the sun.  Then using information about the camera, they could determine the distance between the person in the photo and the location of the camera and the distance of the shadow length on the ground.

So once they learned all the distances they needed, they were able to apply the Pythagorean theorem to determine the photographer  was 5 foot 6 inches tall.  This was the female suspect's height.  So they arrested her, tried her and both she and the male were both convicted of murder.

Most times when you see this type of problem in textbooks, they have you find the height of a tree, a building, or a flagpole.  Honestly, most of my students do not care how tall the flagpole is, all the trees in town are no more than 4 feet tall, and we have very few buildings that are even two story. 

They consider this type of problem as just another boring problem they have to do.  This episode of Forensic Files, add a real life application of something most people study in class.  I admit, when I had to do those types of problems in school, I wondered why I needed to know the height of a building because it should have been in the plans filed with the city.  Flag poles tend to be a set height and why do I need to know it.  Even for trees, I couldn't figure out why I needed to know the height.  After all if a tree gets too tall, the city comes through to chop it down.

This is a situation I plan to use in the classroom to spark interest.  I may have to create a forensics unit with math to make the topic a bit more interesting.  Let me know what you think, I'd love to hear. 

(Rate)(Time) = Distance.

  It is hard to teach rate and distance problems to students who have no real reference to the situation.

My students get problems involving trains, buses, or cars.  The village has all of three pick-up trucks in the village but you are not going to get over 20 mph and you can't go past the dump or airport due to the lack of roads.

The closest buses are in Anchorage but Bethel might have one, I'm not sure.  On the other hand, the only train in the state runs from Anchorage up to Fairbanks and is mostly used by the tourist groups.  Its also so much more expensive than a plane trip.  My students are most familiar with snow machines also known as snow gos, or 4-wheelers which you know as ATV's and they know how long it takes to fly to Bethel or Hooper Bay. 

Even those boat questions where the person is able to go such a speed upstream and another speed returning don't work well where I am.  They use boats but the body of water running through the village is actually a slough which is influenced by ocean tides.  Everything around here is based on when high tide occurs because if you head out at the wrong time, you could end up not having enough water and possibly get stuck when the boat scrapes bottom.

I usually end up rewriting these types of problems to include local places using the same criteria as regular textbook problems but adjusted for my students.  This next year, I am going to look at using animal migration and normal travel speeds to create problems they relate to.

In other words, instead of talking about two trains starting in New York at the same time and going opposite directions, I start the snow goes at Chevak and send them opposite directions because most students regularly travel in the winter to Hooper Bay, Scammon Bay, St. Mary's, or other places within a four to six hour range.  Or I might ask about the Mallard Ducks on their trip south for the winter.

They understand this but when they look at the train problems, most barely know what a train looks like, let alone know how fast one might go and may not be able to locate New York on a map. Many never leave the village or if they do, they usually go to Bethel and possibly Anchorage.

Applying it to train and car problems come after, they've learned to work the problems based on their previous knowledge.  I want to create problems based on bird migration because they understand that birds leave in the fall and return in the spring.  They can tell you the order in which the birds leave, and return.  They may not be able to tell me where they go but they see the migration every year.

They also understand movement patterns for seals, walrus, moose, and whale since there is always excitement as hunters share the information for where they found each.  I've seen seals bob up and down in the water when traveling by boat. 

This is how I work on building a solid foundation for students in these types of problems.  Let me now what you think.  I'd love to hear your ideas on this.

Warm-up

Warm-up

Longhand versus Digital Notes

I read the recent Learning Scientist blog on taking longhand notes versus doing slide annotations and wondered whether longhand or digital is better for notes in general.

I know from earlier reading that writing or learning to write is the kinesthetic piece of learning to read.  The muscle activity helps students learn the letters and how the letters are put together to create words. 

When young children learn to type, it uses the muscles in a different manner and they often find it more difficult to learn to read.  Typing does not provide the same kinesthetic element that longhand does. 

There is research that when a student takes notes using longhand rather than digitally, they tend to score higher on tests.  One big problem with taking digital notes is the easy access to the internet during class.  Most educational facilities offer internet to students as part of the educational experience.  Unfortunately, students often chat with friends, send emails, or surf the web while taking notes so their attention is not fully on the lecture.

One reason for using the internet is based on the fact that typing is often faster than writing notes out by longhand and digital note takers often have to wait for the others to finish writing.  Unfortunately this "multitasking" means students are not focused on the material fully. Consequently, it slows down their ability to comprehend or retain the information.

It has been found that digital note taking decreases retention and recall while shortening a person's attention span.  Remember, yesterday I said a neuroscientist indicated that most people are only focused on a talk or lecture for 50 percent of the time?  This is one reason for a decreased attention span.

On the other hand, writing out notes is slower but it gives the brain time to absorb and store the information while providing the opportunity for deeper engagement.   In addition, research indicates that writing notes by hand uses kinesthetic activity to help the brain encode the information when learning. 

The motor skills involved in handwriting require a person to be more actively involved in learning because of the brain connections which are not found during the passive activity of typing.  The brain is more involved in the physical process of writing.  It uses a more complex thought process because you have to paraphrase the material rather than typing it verbatim. 

Furthermore, it is much easier to write out mathematical equations by hand then try to do the same thing on a laptop.  There are apps out there which allow you to hand write your notes but the app changes your writing into printed form.  I do not know if those allow you to do mathematical expressions or equations.

So this is one good argument for staying old school on notes.  I'm in transit to Hawaii where I'm presenting at the Kamehameha Schools Educational Technology Conference.  I plan to continue publishing every day.  If I learn something cool, I will share it. 

I hope to try out a couple of those convert handwriting to type apps and report back on how they work for math.  This type of app might allow students to take handwritten notes but leave them in a readable form.  My notes resemble chicken scratch so I appreciate that type of app.

Have a good weekend.  Have fun.