Collaborative Ideas For Math

 

We know it is better for student learning if they can collaborate or use peer tutoring to help them learn.  When I got my teaching credentials, I was taught to "show" students how to do the work, so they copied down every step and then provide the assignment.  It's the way I learned and it's the way I default to when teaching.  I know I need to do more collaboration in class but it always seems as if something happens to keep that from happening.

Collaborative learning is also known as cooperative learning.  It allows students to learn with support, encouragement, and often includes learning from peers. I came across suggestions to make the whole process so much easier and which changes everything from teacher centered to student centered so they are doing most of the teaching and learning.

First step is to divide the students into small groups of four to six people.  Give them a whiteboard either physical or digital where the "scribe" or "recorder" will be able to write down all the ideas provided by the group and which every member of the group can see.  It might be jam board, explain everything, or an easel with paper or a large whiteboard.The person doing the recording can only write down what the others say. The other students write down on their paper, exactly what is written on the whiteboard.  Students switch out at the end of each problem.

So the way it works is that scribe one will write a problem up on the board.  The students in the group tell the recorder or scribe how to solve the problem step by step.  Once the problem is solved, everyone writes it down on their paper and the next student becomes the recorder and they repeat it it.  The problem might be a geometric proof, word problems, solving numerical problems, vocabulary, etc.

By having every student work as the facilitator, it empowers them because everyone has to communicate their ideas to the facilitator/scribe.  It also gives students the opportunity to communicate their ideas using mathematical language both verbally and in written form.  In addition, it is good to give students the opportunity to practice vocabulary in a nonthreatening situation, especially if you have students who are English Language Learners.

Furthermore, it allows teachers the opportunities to monitor student understanding by checking out their produced work or watching them in action.  Monitoring is important to make sure that one person is not providing all the answers and that everyone is able to participate and learn.  In fact, it allows students to encourage each other.

It is strongly suggested that this type of collaborative activity be used to practice material after students have undergone direct instruction so they have a better idea of what is going on.  Think of this as practice rather than teacher lead instruction.  It is easy to set up and use.  Let me know what you think, I'd love to hear.  Have a great day.

Millionaire Calculating Machine.

I was watching one of those Pawn Stars shorts on Youtube because it was labeled "Millionaire Calculator".  Of course this sparked my curiosity because I usually associate millionaire calculators with those web based programs that help you figure out when you'll get a million dollars.  This is not what they were discussing. It turns out this is a "Millionaire Calculating Machine" which is the first mechanical machine that actually multiplied.

All previous calculating machines did not really multiply. They added and subtracted but for multiplication, they performed repeated addition but this machine actually multiplied and that was what it was known for.  Now for the story.

In 1893,  the German living in Switzerland, Otto Steiger invented his millionaire calculating machine.  He based it off of an 1878 US patent, and a 1899 French Patent that never went into commercial production. One made it to prove a Spaniard could create something and the other one was more focused on car racing at Le Mans. He was granted a patent in 1895 and shortly there after, it went into production.

This machine differed from others because it used a complex set of cranks, gears, cogs, pins, levers, etc to create a machine that could add, subtract, multiply, and divide.  Where it differed from previous machines is that this machine was set up to read a different metal multiplication table every time the handle was turned.  This means that each turn provided a partial product just like humans do when they remember their times tables.  The machine was capable of carry 10's so you needed one turn to multiply two one digit numbers, a second turn for two digit numbers, a third turn for the three digit number, etc.  In fact, a trained operator could multiply two eight digit numbers in about 7 seconds which was so much faster than any other machines in the past.

Although it was developed to be used in business, scientists found it quite helpful so in a sense it was the first scientific calculator. In addition, governments also liked it.  In addition to being developed in Switzerland, it was also produced there by Hans W Egli. He produced a total of 4,655 machines over a period of 40 years.  The machines came with a hand-operated or electric lever in the basic model, or the more upgraded model that came with a keyboard that could be either hand-operated or run by electricity.

The prices of the machines ran from $475 up to $1,100 in 1924 which is $5,900 to $13,750 in today's dollars.  The price of the low end ones was about the same as a new car. These weighed between 100 and 120 pounds and every machine came with extensive directions and a special brush to keep the machine dust and grit free.  The last machines were produced in 1935 because they were a bit slow when adding long columns of figures.  As improvements happened, the new fully automatic rotary calculators put these machines out of business due to the better speeds.

This machine was the stepping stone that lead to the modern machines.  When I heard about it, I thought it sounded so cool and it would be fun to play with.  I love that it was essentially the first scientific calculator.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 A mangosteen tree produces 500 fruit when they are about 10 years old and up to 2000 fruit per year when they are 30.  What is the average increase in number of fruits each year?

Warm-up

 A mangosteen is a fruit tree that grows anywhere between 20 and 82 feet tall.  If your mangosteen tree is 80 feet tall and 85 years old, how many feet does it grow on average every year.

How Do Bird Flocks Regulate Their Speed?

Its approaching fall here in Alaska.  This means that birds are getting ready to head south to places where it is warmer so they don't freeze to death during the winter. When birds fly as a group or flock, moving in what seems to be unison and its referred to as a murmuration. 

Scientists wondered how individual birds could fly together in a flock maintaining proper distance and speed so they fly as a group.  Consequently, scientists created a mathematical model to explain how birds regulate their speed and position when flying together as a group. At the end, they did more than just create the model, they actually took time to compare the model of flight against videos of real flocks to check the accuracy of their results.

There have been previous studies which looked at how flocks maintain their shape even during sudden directional changes, and how individual birds maintain speed using a linear model but they don't explain how the birds influence each other over the long distances flocks travel. In fact, many of the previous models assumed that birds mimicked the behavior of their neighbor so they all moved at the same speed. In addition, many of the earlier models did not address the individual fluctuations properly.   

To begin with, scientists chose to model the flight behavior of starlings because they have been studied so they had specific data.  For instance, when starlings fly individually, they maintain a speed of between 8 and 18 meters a second but when they fly in a flock, they fly at 12 meters per second.  They've also discovered that no matter how large the flock becomes, the individual birds are able to change their individual speeds to stay at the group speed.

The model they designed ignores small variations in speed while suppressing large ones so they can get a more accurate results.  Specifically, they looked at statistical field theory which is a framework that describes phase transitions.  They decided that all birds have a property referred to as spin, similar to the spin of elementary particles in physics. They concluded that when the birds match their spin with one another, they maintain the total spin of the flock.

They've also figured out that the group turns when a small group began to turn and the information of the turn spreads through the whole flock and it only takes a second or so for the information to make its way to all the birds.  Thus birds do not copy change in direction, they copy the angle of the turn so flocks turn at a constant speed.  The movement is begun by a few birds, and the information spreads through the flock within a second or less, depending on its size. 

In addition, they applied the model different sized flocks ranging in size from 10 to 3,000 members to see what actions the birds took while flying as a flock.  Once they had these theoretical results, they then analyzed individual flight paths within flocks so they'd have trajectories they could compare with the results of their models and the two groups were quite similar.

Although this was only applied to the starlings, it is believed that this model can be applied to other species of birds since many birds seem to exhibit the same movement when flying as a flock.  They hope to apply the model to other types of birds to see if the model holds true.  Let me know what you think, I'd love to hear.  Have a great day.

Mathematics In Geography

 

After stumbling across the Mathematicians Seamounts, I wondered if there were other geographical features with math names.  Instead, I came across the term mathematical geography which is defined as a "branch of geography that deals with the figures and motions of the earth, its seasons and tides, its measurement, and its representations on maps and charts by various methods of projection." according to Merriam-Webster online dictionary. 

So geography looks at the science of the planet itself, relationships such as nature to nature or nature to man or the phenomena in either cultural or natural happenings or occurances.  This means that mathematics is used in regard to the form and shape of the earth, movements within the earth, variables of time and elements of longitude, cartography and map making, climatology, and physiography or physical geography.

Thus I looked into the topic a bit deeper and I found a short abstract listed more ways than I realized.  For instance, Euclidian Geometry is used in surveying small areas such as a field or a house lot while spherical geometry combined with trigonometry to construct map projections.  Furthermore, people have been able to figure out new applications such as using Topology in spacial analysis of networks. As far as networks go,  graph theory has the indices used to describe the different types of networks such as drainage patterns.

Then differential equations are used to study and explain the dynamic processes such as the rock cycle in geomorphology or the study of of the physical features of the earth and their relation to geological structures.  In addition, statistical methods are used to analyze data for regional geography.  Some of these methods are trend surface analysis which uses least squares regression, or factor analysis.  

Of course geography uses a variety of different mathematical models to simplify various problems in geography.  Some examples include the gravity model, simulation models and the Markov chain stochastic model.  Mathematical models can help simulate earthquakes, volcanic eruptions, tsunamis, etc because these all cause disasters and they want to predict where these happen so they can prevent deaths.  

So math is used to find distances between places, the gradient or slope of hills, heights of places, locations given in degrees, minutes, and seconds, perhaps with longitude and latitude depending. 

Now for a small interesting historical fact - This particular secant formula came out of navigation and cartography in the 17th century. The formula - 

is one that many many students struggle with.  It came out of a time when mathematicians and cartographers struggled to understand the Mercator map projection. Originally maps were done on a rectangular grid but Mercator wanted to create a project that preserved both angles and distances.  He figured out a way but was not able to explain it so after a few years, a mathematician explained it using the basic equation which ended up as the above equation.  So here we see how one of the differential equations is used.

This is fascinating how math is used in geography.  Let me know what you think, I'd love to hear.  Have a great day.  

Bookstores, Value Added Tax, Geography, Rounding Off and Math

 I have been doing a ton of traveling this summer.  I just got back from a two week trip to Australia and New Zealand.  Each time I go somewhere, I stumble across something new. Earlier this year, I visited Dubrovnik where I stayed in Old Town.  On the Main Street running north and south stood a bookstore called Algebra.  

The minute I saw the name, my heart went pitter pat and I was excited.  I discovered it was a bookstore with all sorts of books but it was still awesome.  I was actually on a tour of Old Town so I asked the guide if there was anything special about it but he said it was just a book store.  

From what I can tell, it is the oldest bookstore in Old Town and has been located in its current spot for 18 years.  It has a wide selection of books and other things but if you are a book person, you'll be drawn to it.  I'll admit, I just saw its name and that drew me over to check it out more.

I arrived home last night from a very long flight from New Zealand to Alaska.  Had to go from New Zealand to Australia, change planes to Los Angeles before I few to Alaska with stops in San Francisco and Anchorage.  Yes Anchorage is in Alaska but I live further north.  Anyway, Qantas has one of those maps showing where the plane is on it's trip.  As it got closer to Los Angeles, I noticed something called the Mathematicians Seamounts. Mathematicians Seamounts is a group of underwater mountains discovered off the coast of Central America in 1960. Each seamount is named after a different mathematician. The mathematicians chosen are Fourier, Gauss, LaGrange, LaPlace, Leibniz, Lobachevsky, Newton, Pascal, and Poincare.  This is such a cool discovery.

Most every single place I travel to used a Value Added Tax or VAT.  It is included in the price so what is posted on the price tag is exactly what you 'll pay.  There is no tax added at check out but the receipt you are handed shows how much tax you actually paid and the total of the items before tax because if you are a visitor, most countries will refund your tax if you fill out the appropriate paperwork upon departure.  This makes it so much easier to shop because you don't have to determine how much tax you'll need to budget for.

As I said, I visited New Zealand which has the Value Added Tax but their cash registers also do something very interesting.  So they don't have to deal with pennies, the system automatically rounds the price up or down so it ends in a zero.  For instance, if your bill is $19.98, it will round up to 20 and that is what you will be charged and expected to pay.  At first, it seemed a bit strange but then I realized it wasn't bad because I didn't acquire a ton of pennies like I do in the states, if I use cash.

This method is called the Swedish System of rounding. It is usually applied to the total bill so that if it ends in one to four cents is rounded down while if it ends in  six to nice cents is rounded up and anything ending is five is left up to the company to decide how they will handle.  Apparently, it was adopted in New Zealand back around 1990 and has allowed the country to get rid of the smaller one and five cent pieces.  It began in Sweden but has spread to a few other countries.  According to what I've read, it is for cash transactions but I've seen it used when paying by credit card so I see it being used in so many places.

New Zealand Maths has a couple of activities designed to have students practice this type of rounding.  So if you'd like to have your students practice this check the less on here and here.  It is a good way to have students practice rounding in a real life situation.  

I just had to share these odds and ends I've learned this summer.  I hope you found them interesting.  Let me know what you think, I'd love to hear.  I also realize that some of my columns got messed up on when they were published but I kept getting confused when I was trying to set up the automatic publishing.  18 to 20 hour difference can just mess you up.  Have a great week.