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.

Warm-up

 

If the California lives for 770 years and the stand was planted in 1901, how old are they now? In what year will they be 770 years old?

Warm-up

 

If a California Redwood grows about 2.5 feet every year, how tall would this tree be if it was 120 years old?

Post It Notes Part II

 Last time we looked at using physical post it notes in math.  Today, we'll look at a couple more ways to use physical notes before we explore how to use digital ones.  Both have a place in the classroom since the physical ones allow students the opportunity to explore math kinesthetically. 

Post it notes allow students to arrange and rearrange numbers when putting them in order.  It works for fractions, decimals, signed numbers, or even mixed forms.  They don't have to erase, just pull it off and pop it down again. 

They can also be used for tallying numbers, histograms, leaf plots, and so many other data gathering activities.  It translates everything into a visual representation, especially when using one color for each number.  When finding mean, mode, and range, place all the numbers on post it notes so they can be put in order which makes it easier to arrange them for mode and range.  When finding the mean, students type in the numbers in the calculator and move the notes over as they complete each number.  They can also count notes to figure out how many numbers they have.

Now many people use digital set ups which use sticky notes.  They might be used in Google Jamboard as part of the class or as part of Padlet or other digital program. Digital sticky notes can be used by students to identify the parts of the coordinate plane, slopes on a graph, used to identify consistent or disjointed functions, and so many other things.

They can be used as part of a digital parking lot where students are able to record their questions or thoughts,   use it to brainstorm ideas on a topic, summarize concepts or summarize graphs, define vocabulary, or even sharing how they got an answer.  This is a great way to use padlet or other sticky note program.

Digital sticky notes can be used by students to write down what they notice or what they wonder when shown a mathematically based picture or the first part of a three act math task. They can share their thoughts with each other. 

Another way to use the notes for students to give examples of real life situations for slope such as roofs, ramps, mountain road grades, etc. If students are learning about vocabulary words that have both math and general definitions such as product, they can write down the different definitions, of the words and use the words in a proper way such as I sold the product at the store, or the product of seven times six is 42. 

In general, I like to use physical sticky notes for activities that have students doing something physically like ordering fractions while I prefer the digital sticky notes for things that require writing, or communicating their thoughts because the digital makes it easier to save the work. This way I can go back and look at their questions, their thoughts, and their ideas and I am less likely to lose the notes than if I have to keep track of things physically.  

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

Post It Notes Part 1

I love using post it notes in my math classes.  For some things, using the digital ones are great but for other activities, the physical ones are great.  I am going to focus on the physical ones first and digital second.  I love that post it notes came in both the plain variety and ones with graphs printed on them.  I got a bunch from Amazon and my students used the ones with graphs on them.

My students loved the post it notes that had the coordinate planes on them because they could use them as part of their notes when they needed to draw the graph say for logarithmic functions or linear.  The other way they loved using them was as part of their homework when they had to graph anything.  They could just put the sticky note on the page with the problem and draw the graph.  If they made a mistake they could just take the messed up on off and replace it.  It also meant the graph was much more accurate.

I also love using sticky notes as a manipulative to help students learn the processes needed to solve one, two, and multistep equations, or for combining like terms.  For like terms, I give students different colored notes so they can write each term on one but all the x's would be on green, all the x^2 would be blue, xy would be pink, and they would write the term with the plus or minus side.  This way they can rearrange the terms so they are grouped by color since all the same variables have the same color.  This allows students to see visually what constitutes like terms.  I've had students who never moved on from using the colored notes and some who did.  I didn't worry, I just handed out the post-it notes as needed.

Another way I used sticky notes was teaching students to solve one, two, and multi step equations.  If we were working on say 2x + 3 = 9.  I had students write the 2x on one color note and the +3 and 9 on a noter color but they would both be the same color so students would know they were both constants and different from the variables.  Then I'd have a third color to show the process.  So when I showed students to subtract three from both sides, I'd write the -3 on two different notes of the same color and place them under the original equation.  I'd draw a line below that, move the 2x down, talk about the 3 -3 is zero, and 9-3 is 6 and I'd write the 6 on a note on the same color as the +3 and 9.

So now I have the 2x = 6 and using the same color I used for -3, I"d write /2 on two different notes so I could put them below each term resulting in 2x/2 = 6/2. the final line would be the x on the same color as the variable, and the 3 would be on the same color as the +3 and 9.  Then for every problem I used this method for, I'd use the same colors because the colors help students remember and once they completed the problem using this, they could copy the problem onto their paper.  

Other ways to use sticky notes are using them to fill in the real number system graphic organizer, each type of number is assigned a color.  For instance, integers might be orange, irrational numbers might be in pink, decimals in blue, fractions in green, so as the numbers are placed in the chart, the students have the color coding to remind them of the type of numbers.

One thing I see students struggle with is substitution. Doesn't matter if it is a straight substitution of 3x + 2y  with x = 2 and y = 1/2.  You'd write down 3 with space + 2 followed by space. Students would write 2 on one note and 1/2 on the other and place the notes in the blank spaces to get 3 (2) + 2 (1/2) and they can solve it this way.  This can also be done when teaching solving systems of equations using substitution so instead of using numbers, you use equations.  

I'll finish off the physical post-it notes on Friday and address using digital sticky notes in the math class.  Both can be used by in different ways.  Let me know what you think, I'd love to hear.  Have a great day.

 

Geomagic Squares

 

We've all heard of magic squares but not as many have heard of geomagic squares. One person, Lee Swallows, took the idea of magic squares further by investigating geometric magic squares, aka geomagic squares for short. A geomagic square is defined as whose cells contain spacial elements of certain dimension.  Magic squares are considered to be a special case of geomagic squares where all elements are one dimensional.

Think of it this way.  The numbers in the magic square represent the segment lengths in a straight line that add up to a total distance. In addition, a geomatic square might be made up of 2 dimensional areas that when put together create the same area with the same shape such as squares.

The same applies to if the square is made up of individual volumes that when combined create the same 3 dimensional shape with the same total volume. 

In regard to two dimensional shapes, they might be geometric shapes both regular and irregular or sections of a circle that when combined in any direction will form the same circle.  The circle ones will be based on the standard magic square.  Think of it this way.  360 /15 = 24 degrees.  The 15 is from the total that each row, column, or diagonal adds up to.  Thus if you look at the magic square, the first row first cell 

would have part of a circle that is 8 x 24 or 192 degrees, the second cell would show one segment or one that was 24 degrees while the last would be 6 x 24 or 144 degrees.  If you add them all up, you get 192 + 24 + 144 which adds up to 360 degrees,  If you repeat this for for the second and third rows, you get a full complement of segments that do add up to full circles.  

There is a general formula that can be used to describe magic squares is :

Now we can use it with numbers such as C=5, A = 3 and B = 1 you end up with the magic square as referenced above but what if you assigned A, B, and C to shapes.  Think of A = a small rectangle.  The C represents a square with sides equal to the length of A and B is a semicircle of diameter equal to the length of A, you can use those shapes to form a visual representation of magic squares. The idea is that you may have to rotate the shapes to get them to match up to form the shape.  

Now there are so many applications of this basic idea that I don't have the space to go into it, so you might want to read this article by Lee Swallows.  I was impressed with the different ways one could represent the basic magic square referenced above.  Read the article then check out the gallery at  this site by Lee Swallows to see what all one can do with the geomagic squares.

Warm up

What is the population of New Zealand if the population is 5 million people spread over 103,484 square miles? 

Warm-up

 This New Zealand tree can live to a thousand years. If each ring is about 6 to 8 months growth, use the formula to find the number of rings for an 80 year old tree. The formula for age is to divide the number of rings by 2 and add 20.

Algebra Touch

 

Algebra Touch is an app that has been around for a few years but the people who originally released it, have released an updated version that has some really nice additions.  In case you're never checked this app out, it is different than many other math apps because it focuses on teaching students how to do the process by actually having them do it.

Algebra Touch has the student use their fingers to combine terms, move terms around, or other movement so they actually do the process.  The old version had this facet but something new has been added which I really like.  I'll tell you about that later on.

When you open the app, you get a list of 8 topics that can be explored and practiced. Topics are divided into beginner, intermediate and advanced.  The beginning topics are like terms and order of operations, intermediate cover factorization and elimination, while advanced is equations, distributions, exponents, and logarithms.  Each topic has a certain number of subtopics to practice and at the end of the topic, there is a challenge practice for students to try.

 The topic on the main page, tells you how many subtopics it has.  When  you click on a topic, the submenu comes up so you know what you will be practicing. In addition, there is an introduction to give you a short description of what is happening.  Then you work work your way through each subtopic.  At the bottom of each subtopic is a create your own choice where students can input their own problems.  When you type in a problem of the same type, it allows you to solve that problem.  This means students can type in assigned problems, work through them, copy down the steps to show their work, and they are done.  

When you click on a sub topic, in this case I went to the like terms one, 

and chose the first subtopic.  I got a demonstration screen to show me how to do this and then it went to the first problem.  to do this I would place my finger on the -4a and move it over by the 19a so they are next to each other.  To combine the terms, you touch the + button and it combines the 19a + -4a to make 15a.  Then touch the + between the 12 and -10 to get 2 so the answer is 15a + 2.  It doesn't matter whether the terms with the variables are first or last.  In fact, the app does not make you put the variables first.

Now if the student has trouble figuring out how to do this, every problem has a video link at the bottom of the page.  If you look above, that little tab that say something like 0 of 6 which tells you how many problems in the section and how many have been done.  If you click on that tab, it rises and has an embedded link to a video so the student has video help available for every single problem including those in the challenge but not the do it yourself problems.

As far as I can tell this app is free and is available for the iPad and iPhone.  I love playing with it myself.  I do like that even if the students choose to use the create your own option for problems, they still have to work them out and that is awesome.  Give it a look and check it out.  Let me know what you think, I'd love to hear.  Have a great day.

Magic Squares and Observations.

If you look at the instructions for how to do a magic square they have a specific progression on how to fill them in. Today, I"m only looking at the 3 x 3 grids but I plan to look to see if these observations for a 3 x 3 work for larger square and could lead to a interesting activity.  I used numbers for this but it could be done by hand or by excel.  

To create a 3 x 3 while using the digits of 1 to 9, start by putting the 1 in the middle cell of the top row.  

The next digit 2, is located one pace up and over but that puts you outside the grid so you end up placing the 2 in the bottom right most square.  The digit 3 is then placed right one and up one so again, you are off the grid and end up placing it in the first cell of the second row.  

The digit 4 is one cell down, just below the 3. 5 is then one cell to the right and one up and the 6 is also one right and one up so the 4, 5, 6 are on a diagonal from lowest left to highest right. To get to 7, you move one cell down.

To place the digit 8, you go right one cell and up one so you have to move to the top row, first cell.  The digit 9 ends up being placed in the last open cell between the 4 and the 2.

The next step would be to have students add up all the rows, columns, and diagonals, to make sure they all add up to 15.  If you are using numbers or excel, have the students create the math for the cells. 

Once this is set up, for all the cells, students can see they all add up to 15.

Now for the fun part, ask students if you rotate the magic square 90 degrees, 180 degrees, or 270 degrees, will it still work out properly?  I did it and yes, the magic square still adds up to 15 in all directions for all those rotations.

                                               

We all know it is but many students wouldn't and it makes a nice extension for the activity and it brings in the idea of rotation in an area other than geometry.  Let me know what you think, I'd love to hear.  Have a great day.

One Less Real Life Grid Example

 

As I write this, I am in my hotel room located in Sydney, Australia enjoying a nice laid back day.  I spent some of the day online looking for some tours so I could see more of town.  Many of the tours do not pick you up at a hotel so I was on google maps trying to see if the point of departure or return was within walking distance of my accommodations or if there was another hotel near mine so I don't have far to walk. 

Google maps are so different from their predecessors because they automatically identify where you are and you input your location.  It gives routing from where you are to where you want to go so you could go by car, bus, or walking.  In the old days, we'd use a variety of maps which used grids.  You might check out the map section in the phone book, a map, or those books used by real estate agents.  

All of these use the same process.  You look the address up in the index, go to the correct page and the focus on the part of the grid containing the street.  It is at this point, you actually start your search. For instance, you might look up the road in the phone book map section and it sends you to page 45 to look for section I-5 and that was all you did to find the location.  

I mentioned a book because a friend of mine who lived in Los Angeles had this huge book used by real estate agents, listing every road, every street, every anything so if she had to go to a new place, she could find her way there.  Every page in the book was part of a larger grid, so if you took every page out and put them together, you'd get a huge map but it didn't suggest routes at all.  Instead you were expected to flip pages forwards or backwards looking for the previous or next part of the grid, until you figured it out.

Since I've never really lived in large places, so I could get by using the map section in the phone book for local things and a regular map when I needed to cover a longer distance since it contained enough info but maps did have grids to help narrow the search on a map.  

I've been using Google Maps quite a lot recently and just the other day I realized that google doesn't use grids at all.  I suspect this is because google is set up to show your location and you choose the other location and voila, you see it all.  There is no need to search through the phone book, a map book, or a paper map because it's done for you.

I'm a bit sad about this because students no longer have to learn to read the usual maps with their grid systems.  In fact, I'm not sure many of our high school students know how to read a map or even identify a map.  It is a skill that is no longer needed but that means it is one less real life examples I've used in the past but I really can't use it anymore. 

It's like watching one of the original Superman episodes where Clark Kent would dash into a telephone booth to change into his alter ego Superman.  By the 1970's or so, the full length telephone booth had morphed into those new ones with the phone attached to a half sized one and then by the time cell phones took over, no more phone booths of any sort.  In fact, we can't even talk about cell phones or rental cars with the base rate plus so much per text or call or mile so those are out because we don't operate that way anymore.  

I'm still working on finding a replacement example but it's slow.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If one cup of heavy whipping cream produces two cups of whipped cream, what is the percent increase in volume?

Warm-up

 

If you add two tablespoons of cream to a cup measure and fill the cup with skim milk, you can make your own whole milk. What percent of the cup is cream. (Hint: one cup has 16 tablespoons in it.)

Is The Golden Ratio Test For Beauty Accurate

 I know I've looked at this topic before but there is more evidence that this rule is not valid.   The last time, I provided the logic used for why it worked but this time, I'm focusing on the argument for why it isn't true. As humans we always want explanations for every thing especially if we use ideas or theories we are comfortable with.

Many years ago, a cosmetic surgeon stated that Amber Heard had one of the most beautiful faces based on "The Golden Ratio Test".  The idea is that the closer the facial proportions match up to the golden ratio, the closer one is to "perfect" beauty. 

As we know, the golden ratio was discovered a very long time ago by the Pythagoreans who named it "The Divine Proportion".  The ratio is equal to 1.618. The Pythagoreans were made up of mathematicians who believed that certain numbers had mystical, philosophical, or ethical significance.  Now where the idea of beauty and the golden ratio came from, came from Greece, the Renaissance, and even more modern times.

Plato proposed that our world is an imperfect copy of the real realm of truth since we see a perfect triangle exists in truth but in our world, a perfect triangle is not created naturally.  He went on to say we can glimpse the perfection through the use of logic or by creating symmetry.  Fast forward to 1509 when a famous mathematician of the time published trilogy on the Golden Ratio complete with illustrations by Da Vinci. One point he promoted was that human bodies should match certain mathematical ratios.  

Then in the late 1800's another mathematician expanded these ideas into multiple books but in his last book, he suggested that the most beautiful and fundamental ratios must be related to the golden ratio and it can be seen in the body, nature, art, music, and architecture.  This lead to the belief that both Ancient Greek art and architecture used the golden ratio, therefore they were beautiful.  

Later on, someone actually looked at this belief.  It appears that no mention of the golden ratio appears in anything outside of math and numerology in Ancient Greece. As a matter of fact, phi is seldom seen in Ancient Greek art or architecture. In fact, the Parthenon was voted the most beautiful building in Athens in 2014 because it had phi as part of it's ratio but in reality, phi does not show up there.

In 2002, a cosmetic surgeon claimed the golden ratio can be used to determine the most beautiful people because the ideal face would have a mouth that was phi wider than the most.  The surgeon then created a mask to show visually, the application of the golden ratio for cosmetic surgeons and orthodontists. In addition, he claimed the mask could be used to find perfect beauty and it lead to "The Golden Ratio Test."

He measured the faces of actors and models to provide data to support his theories. Since the test became so popular, it lead to others using the test to define beauty and others who developed apps so everyone could easily see how they matched the definition of perfect beauty.  Fortunately, someone actually looked at his data and calculations to determine the test's accuracy.  His limited data did not include people of sub Saharan, East Asian, or South Indians but focused on those from Europe who were models. Even then, a look his calculations indicated that the correlations between the ratios of the faces he used and the golden ratio itself was statistically insignificant. 

So there you are, where the test came from and it's validity.  I will get back to normal on Monday once I'm where I'm headed and had a chance to catch up on the huge time change.  Let me know what you think, I'd love to hear.  Have a great week.

Hotel Bills

 I know I didn't publish anything yesterday.  I spend the day, down in the Los Angeles Fabric District in downtown LA.  It is right next to the garment district and the two intermingle.  Today, I checked out of my  hotel and I just sat down to write this.  It is based on the checkout bill.

The rate quoted was $147 per night which is not bad for a nice large room at the Marriott.  The thing is, this rate did not include any taxes because each city and state has the different rates and different taxes.  

The hotel was in Woodland Hills, California so they have one set of taxes.  I paid $20.58 for the room tax, $2.94 for a tax labeled a "TMD Tax" which is some sort of California tax and $0.22 for a California tourist tax.  So all in all I paid  $170.74.  This is a great exercise in figuring out the tax rates for room tax and the other taxes.  

A few years ago, I stayed at this same hotel and they included an energy tax to cover the rolling brownouts  the state was experiencing.  So that was even more.  Another time, I stayed at a place classified as a resort in San Diego.  When I went to check out they had the usual room tax, the TMD tax, and the California tourist tax but then they had a resort tax and a tax on the resort tax.  I always look at my bill to check things out,  When I saw the tax I didn't recognize, I asked them about it and the front desk staff had to run to the accounting department to get an answer.  It took them about 20 minutes to get an. answer. 

One way to check tax rates out is to look at various booking sites.  For instance, Booking.com tells you the base rate and that it excludes the taxes while giving the rate.  For instance, I looked at a hotel near LAX with a daily rate of $215 which includes breakfast but not the 16.2 percent so students can use that to calculate the actual daily rate including tax.  If you reserve the room, it will give you the absolute total you'll pay including all taxes.  The other thing is that this site will also tell you if there are any "cleaning fees" that will be included in the daily rate.

Another site, Hotels.com lists the hotel, the daily rate and the total rate on the page that lists the price. I chose a hotel and they had a link to the price break down so you'd know the exact amount for the daily rate, the taxes and fees, and the final price but it doesn't give percentages. In addition, if you want the privilege to be able to cancel later, you have to pay an extra $10 for the ability whereas many of the places with Booking.com, you have that given freely.  

Expedia.com handles the prices the same way that Hotels.com but has free cancellation up to a certain point which is nice.  Then there is Kayak.com which lists several different websites and their prices.  There is a wide variety of prices for the room I looked at but some services had free cancellation but each had the percent of tax listed as a percent.  If you take the room, it gives the actual amount of tax you will be paying.  

These different sites give students a chance to learn how to read them, figure out what the final price is and look for things like "Is there breakfast included in the final price?"  Sometimes I choose a place that serves breakfast because the price different isn't that great and it works well but if the room with breakfast is like $30.00 more per day I might choose the other option so I don't have to pay as much.  There are so many things one can have students do with these sights including a compare and contrast activity that requires them to make conclusions and use higher order thinking skills.

I tend to use Booking.com so much because I know if how the taxes break down, cleaning charges, resort fees, etc and I know the final price.  Let me know what you think, I'd love to hear.