Tis Almost Easter

Good Friday

 

Out here in the village, the week between Palm Sunday and Easter Sunday is important regardless of the denomination.  Although we can do things on Monday, Tuesday or Wednesday evenings, we do not plan anything for Thursday to Sunday or its considered disrespectful to the culture and to the churches in town.

I wish everyone who celebrates this weekend has a wonderful time.

Exponential and Decay Functions

  At some point, we teach students about exponential and decay functions.  The goto examples for exponential functions is unlimited population growth be it people or bacteria but what are some other possibilities for exponential growth?

Exponential growth formulas are used when something grows at a certain percentage every year while exponential decay are used for situations where something decreases a specific percentage every year.

One that is fairly new, is the growth of social networks such as Facebook.  Since 2004, these social networks have experienced an exponential growth.  Facebook itself has gone from one million members in 2004 to over a billion in 2010.  That means if Facebook were a country it would be the 3rd largest country in the world.

Other companies with exponential growth include Uber transportation and AirBnB companies.  Ideas which just took off.  For instance Uber has grown from $200,000 in 2009 to $1,200,000,000 in 2014 while AirBnB has only grown from 7.2 mill.  Image giving students these figures and letting them figure out the formula?  That could be fun because it students are looking at real companies.

The United States Gross Domestic Product per capita has grown at a steady rate of  two percent per year for almost two centuries.   Another area with exponential growth is continuous interest on money investments.  Its always interesting to compare continuous interest with simple interest to see how much of a difference there really is when all is said and done.  My students have been surprised by that.

One thing I found interesting that is an exponential growth is the increase of computing power.  its said by 2030, the computing power of a computer will equal that of the human brain. Even consumption of fuel has exponentially grown over the past couple of centuries.

Then if you look at the negative exponential functions often referred to as decay there is the old radio active decay situation but many of my students do not relate to that.  Instead bring up how coroners use negative exponential functions to determine the time of death using a specific formula.  

One of these decay situations involves a bill that a politician tried to pass which would force restaurants to decrease the amount of salt used in cooking by two and a half percent each year.  The politician tried to pass this law because people have been consuming too much salt but the law did not pass.

In addition, the decay formula is used to calculate the amount of formula on a loan whose balance is decreasing.  If students ever look at mortgage payments They will notice that most of the payment at first is to pay off the interest while at the end it changes to be mostly principal. 

Further more, the value of cars is considered to be an exponential decay as it ages because its value decreases the longer you own the car. 

So many situations other than population and half life situations.  I'd love to teach this next year by having students choose a specific situation, find the formula for the situation and then create a short video to present their information.

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

Why Teach Reading and Writing in Math

  When I first started teaching here, ten years ago, there was a program in place designed to teach reading and writing across the curriculum but as superintends and principals have changed over the intervening years mandates have come and gone.

At the moment, this is not a priority which is a shame because when we have the mandate to teach reading and writing across the curriculum and we offer a specialized reading class for those who struggle.

So why should be teach reading and writing in math class when it should be done in English or Language Arts.  While hands on activities are great for building prior knowledge, creating interest, but it is the reading and writing that help a student analyze, comprehend, and communicate mathematical ideas.

It is well known that the more a student knows about a topic the better they understand the material as they read it. In addition, the specific reading skills needed in mathematics are the same ones used in reading so it is better to teach the skills in both subjects so students see the connection.  These skills include comparing and contrasting, inferring, predicting, and cause and effect and are important in both topics.

In regard to writing, students clarify their understanding, improve their communication ability, organize both their ideas and thoughts into a more coherent form, and create better conclusions.  The only way to accomplish this is to have students practice writing. 

Reading and writing can incorporated by using trade books, texts, and fiction books with a math theme because they provide a look at concepts in a fun way.  Most of the fictional books I've read are picture books designed for younger readers, these same ones could be read by high school students before analyzing the math and writing a summary of the story or concept.  

I am currently taking a class on integrating film making and animation into the classroom.  Why not have students read the fictional books and then create a short one shot film discussing the concept covered, or a review of the book, or even their favorite line. They would have to write the script before recording it.

Other ways to increase the amount of writing in class is to require students to record their thoughts in a mathematical journal.  They could record their thoughts on solving problems, their weaknesses or strengths, questions, or their opinions on some mathematical topic perhaps taken from real life, write a letter on an issue that includes data from the census bureau, or other reputable data source. 

It would be nice if the English department and math departments could work together to plan the teaching of certain skills in both classes at the same time.  If cause and effect is being taught in English, do it in math at the same time so students practice it in both subjects.  We keep saying skills need to be taught across the curriculum yet most schools do not.

Maybe we need to change the way we think about the way we teach the topic.  Let me know what you think.  I'd love to hear.

The Language of Mathematics or lack of it.

We all talk about how important it is for students to know the vocabulary of mathematics but how often do we as teachers really use mathematical language as much as we can?

I use the normal vocabulary but this past Friday, during a game of Kahoot, I realized I didn't use it as often as I could.  You are probably wondering what I'm referring to.

The topic for the Kahoot game was solving one step equations.  I read the equations the way I usually do such as "x + 2 = 4" is read as X plus two equals four.  I realized I wasn't using all the mathematical vocabulary I could.

I immediately switched to reading it as "a number increased by two gives four".  Instead of teaching students to write the equation in verbal form for one or two weeks a year before ignoring it and reverting to the way we normally read equations.  I realized I need to spend the whole year using the verbal form of any equation so they build a better connection otherwise, they will always have that disconnect.  They will always see the two forms as totally unrelated.

This is especially true for English Language Learners so they become more fluent.  The thing we forget as math teachers is that language is made up of more than words.  It is made up of the words placed in the correct order so the idea is fully communicated both in written and spoken form.  Many teachers, including myself, forget that so we do not spend the time connecting the two forms.

In addition, I've started putting an equations such as 2x = 8 for students to create their own word problems.  This is an open ended activity for students to build vocabulary, begin to connect equations as a written representations of verbal situations.   A very important skill for today's world.  I love my students are coming up with different situations for the same equation. 

I feel this helps them improve their language and mathematical language because it improves their ability to express their analysis, interpreting data and ideas and them communicate that information in a way others can understand. 

This idea just mine.  I'm having trouble finding research on the topic.  When I've looked up information on the topic, I mostly find information on teaching it when you reach that point in the curriculum where students learn how to go from one form to the other and back again but not as much on relating the two throughout the year in other circumstances. 

I did find some great information on reading and writing in math which I will share with you tomorrow.  Let me know what you think, I would love to hear.

Parabolic Art

Example of completed parabolic art.

 The school I work at does not offer drawing in the high school. The only art offered is native crafts but too many of my students want to draw.  They draw whenever they can.

So I decided that during the final quarter of the year, Wednesdays would be "Art Day" but it would be mathematically based art.  I looked around and found Parabolic Art.  The big difference between what I'm having the kids do and what is found on the internet is I have my students draw whatever they want in the center just like the big one at the top.

The first step is to create a square.  I told the kids 6 inches by 6 inches.  Then have them mark dots every 1/4th of an inch.  Already the students are

practicing measuring in small increments.  My students have trouble using a ruler and this is going to give them practice.

After the dots are marked, they will start drawing straight lines from the corner to the first dot as seen in the picture next to this paragraph.

They will connect the next dot with the dot above the end side.  Every time, they will draw a line that starts further left or right and ends further up until they've reached the top.

When they are done, they will see a beautifully curved line with a unique perspective to the resulting squares.  This is where many people stop because its finished the purpose of the lesson.

I went had the students repeat the process through three more times so they had a parabolic curve on each of the four sides.  When completed, there was an empty space in the center.  This is the area, I gave the students to draw anything they wanted. 

One student completed theirs and they loved the process.  The others are having a great time creating the lines using a different color for each section.  It looks like two eyes crossed at 90 degree angles.

You don't have to have all four corners colored, you can have them do two and have a bigger space.  Some of the students did this back in elementary school using thread and cloth.

This picture was created using the x and y axis for the coordinate plane using the same technique. 

I subdivided the axis into smaller units and ran straight lines from one end to the next, just like I did in the rectangular example above.

I colored in some of the boxes to make it more interesting and then drew some flowers around the creation.  I love the open space so students can make the art totally theirs. 

There are other shapes that I am playing with at home right now.  I promised Kiki, I would write something up on this. 

When I've figured out more designs and shapes, I'll share them.  In addition, next month, I'll be working with the second grade to create pi based cityscapes.  There is no reason, we can't throw in mathematically based art occasionally for students who would rather create art than solve equations.

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

Warm-up

Warm-up

Knots Vs Miles?

  In most anything people deal with daily, speed is given usually in miles per hour.  We have speed limits in mph, we talk about running in mph, bikes, motorcycles, etc everything is in mph. 

The only two forms of transportation that do not use miles per hour are marine vessels such as ships and airplanes. They record speed in knots per hour.

There is an interesting reason why sailors use knots.  Its because they didn't have any other way of figuring distance.  One knot is a nautical mile which is 1.852 kilometers.  The knot comes from tying knots in a really long rope every 14.4 meters.  They would tie one end of the rope to the ships stern while they'd tie a wooden board to the other end.  The board would be dropped into the water.  A sailor has a 30 second timer and would watch it, while the other sailor counted knots that slipped through is figure.  They took regular readings to tell how fast they were going.  Over time, they declared that one nautical mile is equal to one minute of latitude or 1.15 land miles per hour. 

Now we know why they use knots for maritime travel but why is wind speed in knots rather than miles per hour?  Wind is made of air and is classified as a fluid just like water so both are measured in knots.  In addition, wind actually pulses rather than flowing steadily so knots give a more accurate reading. 

From what I can tell, many airplanes list speed in both knots and miles per hour depending on when it was built.  It also appears that airspeed is listed  in knots or nautical miles per hour while ground speed is in miles per hour.  Air traffic controllers expect speeds to be given in knots when a pilot calls in for information and the automatic weather information radio broad casts give wind speed in knots.

Now if you'd like to include some activities on this topic, here are two suggestions.

1. NASA has an activity which requires students to convert from knots to miles per hour and back again.  This is a nice way to introduce the converstion.

2. This site uses information on the USS Mauretania from the turn of the 20th century when it cossed the Atlantic in under 5 days.  The activity has students calculate how much income was generated through ticket sales, calculate the average speed in knots and miles per hour, statute miles, and kilometers.  Lots of nice conversions.

Why did I choose this topic today?  I wondered why there is a difference in speed choices for marine and land vehicles.  I thought it was interesting.  Have a great day.  Let me now what you think, I'd love to hear.

Dinosaurs and Scale Models

  The other day while researching information on using scale model drawings for sculptures, I hit the jackpot when I stumbled across a website filled with videos and activities for scale models.

I am saying not 3, not 4, not 5, but 17 activities.  The over all title of the collection is "Scale City: The road to proportional reasoning" thanks to PBS and their Think Math.

Scale City with 8 video's on different aspects of scale models and accompanying interactive activities which has the students practice the skill introduced in the video.  The first topic is one dimensional scaling in the real world and it uses full scale models of dinosaurs.  The activity has students comparing a boy to a T-Rex in terms of size.  There are four questions, two hints, and once done the activity is graded so there is immediate feedback.

The second activity focuses on similar figures in the real world.  The video examines the world's largest bat at the Louisville Slugger Museum and Factory. They learn about what it takes to make a bat and are asked to compare the size of a regular bat to the one in front of the factory.  The interactive has them use shadows to find the height of an unknown object.

The third topic examines outdoor murals to see how measurement and proportional reasoning is used by artists to scale up the piece of art from a scale model to the finished product.  The interactive activity examines the two dimensional scaling up of rectangles.  The fourth looks at scale models in real life such as model trains, doll houses, and architectural models while the activity encourages students to explore three dimensional modeling.

The fifth explores scaling up or scaling down recipes and how the change in the radius of a circle changes its area.  The sixth video focuses on the relationship between the size of an items shadow and its distance from a light source while the seventh introduces students to the idea that the length of a pipe or string has a proportional relationship to the sound produced.  The eighth one focuses on proportional relationships in real life while the last is a review.  

It is wonderful the way each video has an accompanying activity.  In addition to the interactive online activity, each video has a lesson plan which gives times for the introduction, the movie, and a small group activity before students are assigned the computer based activity.  Furthermore, there is a small connections section which provides links to other sites.

This is a collection of videos and activities designed for grades 6 to 7 but could easily be used with older students, especially ELL students.  I like it and plan to use it later on in the year when things slow down.  It is great to see something that has lots of real life examples for each aspect of the topic.

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

Flashcards and Math

  I'm looking at the use of flashcards in today's entry because I work with teachers who still hand students packs of flash cards to work on learning their multiplication facts.  I'd love to use them in high school if they are indeed an effective way of learning.

After reading up on the topic, flashcards can indeed be effective because they provide active recall.  When you look at the front of the card, your brain searches for the information.  Active recall creates stronger neural connections and flash cards allow for easy repetition. 

Flashcards have you use your meta-cognition because you have to determine if your solution is correct. It makes you do self-reflection as you compare your answer with the flash card and it helps place that knowledge deeper into your memory.  Finally, flashcards provide confidence-based repetition which helps improve memory. 

When it comes to making flashcards, there are ways to create the cards so they are more effective than just the standard drill and kill ones.  There are eight things you can do to improve the use flashcards.

1. Make your own flash cards because it helps your brain intake the new information while starting to build strong neural pathways to recall the information later.  When you create your own flash cards, you are putting it in your own words.  When you use pre-made cards, you skip these steps.

2.  Add pictures to the flashcards because most people remember pictures better than words.  Do not replace words with pictures because that will not set things up for optimum learning.  It must be a combination of words and imagery. The written part can be a short descriptive sentence or a single word.

3. Uses mnemonics or sayings when you can otherwise select acronyms, rhymes, or associative images to help you remember.  In general, the wilder and crazier the association, the better you will remember the information.  Since you are making the cards for you, the association only matters to you.

4.  Do not write more than one fact per card.  When you place several facts on the answer side, you trigger the illusion of competence rather than actually being competent.  Our brains are good at recognizing something we've seen before but its not the same as actual recall.  Recall occurs without without a clue. 

5.  Be sure to break the more complex topics into multiple questions.  For instance if you want to look at the groupings of the periodic table, you need one question for each group.  In math, you might ask yourself about the three possible answers but do it using three specific questions, each on a different card, otherwise you'll fall into the illusion of competence.

6.  Be sure to answer questions out loud because you must organize your thoughts in the process.  If you have a friend, let them ask the question while you answer out loud, otherwise you could record your answers to play back while you compare.  If you answer silently, it only continues the illusion of competence.

7. Now for the big idea.  Study both the material from both sides of the cards.  Run through them testing yourself via the front of the cards, then run through them using the back side for clues.  This helps build neural pathways that run both ways rather than the traditional one direction.

8. Furthermore, do not use flashcards as the only method of reviewing material.  Make it only one of many such as writing down what you remember in your own words, write a quiz, take a test made by someone else, make mind maps or Venn diagrams, or work lots of practice problems.

So if you are a person who likes flashcards, these are some suggestions to keep in mind.  I didn't know some of this before I went looking because when I was younger, you went through flash cards over and over so you felt confident but when the test came, you promptly forgot it all.

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

Optimization and Spreadsheets.

  Optimization is looking at something to find the best solution.  It is used in manufacturing, production, inventory control, transportation, scheduling, networks, finance, engineering, mechanics, economics, control engineering, marketing, and policy modeling.  Mathematical optimization is an applied form of mathematics used in so many different ways.

Unfortunately, most times, optimization activities use calculus but it is possible to expose students to trying without students knowing calculus.  The cool thing about using spreadsheets is that it gives students a visual representation.

Optimization has its own set of vocabulary from the objective function which is what you are trying to maximize or minimize, variables which represent those items you can control, and constraints that control the size of the variable.  An example of the vocabulary applied to a problem would be from the football coaches point of view.  He wants his players to gain the most yardage possible each time so that is his objective function.  The factors such as practice time for weight lifting, running, ball protection, etc.  would each be represented by one variable.  The constraints involve the total time available to practice, the number of acceptable fumbles, the number of times a player is tackled, etc.

There are several types of problems easily used in the Algebra classroom with a spreadsheet and minimal knowledge of spreadsheet use.

1.  Maximizing the volume of an open box.  Start by giving students an 18 inch by 18 inch piece of paper.  Have them cut small squares x by x out of each corner.  The value of x is from 1 inch to 8 inches. Once the squares are cut out, students assemble the boxes so they can see the final shape of each. With some discussion they should discover the formula is 18-2x = length = width and x is the height.  Students then fill out a spreadsheet to calculate the volume form the height, length, and depth of each box.  Once they have the results for the volume of each box on the spreadsheet, students create a graph showing the curve of results. 

A variation of this problem would be to create an optimization spreadsheet for the constraints of the post office.  They have a maximum of 108 inches for height and girth so what is the best shipping size of boxes.  That one students out here understand because everything has to come in by air unless it is too big and too heavy in which case it has to be shipped by barge or by air freight or bypass mail.

2.  Maximizing profit, an important optimization if someone is planning to open their own business.  A shoe company makes shoes for $15 per pair and sells them for $45 per pair minus a 5 cent discount for every pair a company buys.  The formula is (45-.05x)x = income and $15x is the cost of making each pair.  The 45-.05x is the cost of the shoes while x represents the number of shoes.  There will be columns for number of pairs, price per pair, cost of shoes, income and profit.  Students will discover that buying 300 pairs is the best buy but if they sell more than 600 pairs, they will no longer make any profit.




For other ideas this paper has some in depth information for optimizing investment among several investments.  The paper includes the constraints, the variables, etc, everything needed to run a scenario.  In addition it also provides all the information to to an optimization of a network system running from Los Angeles, CA to Amarillo, TX.  These are a bit more complex but they can be done on a spreadsheet.




Here are some really nice problems for students to learn about optimization without having to know all the calculus necessary to solve them.  Have fun and enjoy.  Let me know what you think, I'd love to hear.

Hunger Games

Hunger games is the name of a wildly popular book that was later made into a book.  Yes its been out a few years but its still one of those movies that can grab a students attention and keep it.

The basic story line for the story is 12 districts send a boy and a girl to compete in the nationally televised "Hunger Games". 

They fight it out until only one survives.  This time, the heroine must weigh love against survival.
The boy and girl who are selected to compete are randomly chosen during a ceremony from the 12 to 18 year old segment of the population.

A few years ago, the high school reading teacher used it in her class.  I didn't realize it at the time but it would have made a spectacular cross curricular unit between reading, math, social studies, and science.  From a mathematical point of view, Hunger Games provides a wonderful application of probability to a situation that students might relate to.

The selection process is actually a lottery in which the more times a name appears, the better the chance a person has of being selected but if their name has been selected, it is removed from the pot because they are either dead or won.  The reason a child has their name appear more than once is that it is put in again every year so at age 12, the name is entered for the first time, at 13 the name is put in again and it continues so at the age of 18 a person has multiple chances of being selected.

The above is a perfect example of arithmetic progression.  This means that when they are 12, the odds of being selected is low but by the time they are 18, the odds have increased tremendously. Wired has a beautiful article showing an example of how it works in this situation. It is well written and the example is easy to understand.  This contains a summary of the original one which has more detail and mathematical equations to use if students are interested.

The article points out that some children can have their names appear more if the family volunteers to have the name entered additional times for higher allotments of food, etc. One way to accomplish this is to create a random generator or even use a coin to choose a 1 or 0 representing the yes or no for another entry for more food.  The NCTM published an article on a Hunger Games activity in their Middle School Magazine Volume 17, issue 7 - the March 2012 publication that has the random generator.

In addition, this article looks at the probability of winning the event itself.  If you think about it, 2 contenders from 12 districts means there are 24 young adults trying to be the one winner.  That means each person as a chance of 23/24 of loosing the event.  Those are not good odds.

The other mathematical aspect is game theory or the theory of making independent choices based on the decisions of others.  The most famous of this type of theory is the prisoners dilemma.  The prisoners dilemma goes as follows - There are two suspects being questioned for a major crime but there is not enough evidence to arrest them on that but there is enough to get them on a minor crime. So the police offer them a choice. 

The one who confesses to the crime will be released while the other one will be convicted and spend 15 years in jail.  If neither confesses, they will be convicted of the minor crime and spend one year in jail. If they both confess, they will only spend 5 years instead of 15 in jail.

Although the best move is to stay quiet, the prisoners are under a lot of pressure to confess.  In the book and movie, contestants band together to form alliances or groups at the beginning.  But how much can they trust each other if they know the object is to kill the others.

I found the idea of creating a cross curricular unit quite appealing.  I don't know if we can do it but I'm going to propose it for next year.  Let me know what you think, I'd love to hear.

Warm-up

Happy St. Patrick's Day

Mathematical Sculptures

  I was trying to find information on the mathematics used by sculptors and instead of finding that, I found sculptures based on mathematics.   I didn't know how many artists use mathematics to create their art or sculptures.

In today's world, mathematics is an important component in both animation and digital graphics.  In addition, certain artists are using mathematics to create sculptures.

Ricardo Zalaya Baez wrote a proposal for classifying mathematical sculptures based on the type of materials and mathematical properties.  He divides the art into geometrical sculpture, sculpture with concepts of calculus, sculptures with algebraic concepts, topological sculptures and sculptures with different mathematical concepts.

Through out the paper, he includes photographs of examples of each type of mathematical sculptures and their creators.  Here are some who were chosen.

Helaman Ferguson who has a doctorate in mathematics but is an artist who uses mathematics to create his sculptures.   One of his creations is Umbilic Torus SC is created using a computer program designed to direct a robot to care 144 different sandstone pieces which were later cast in bronze.   Many of his other sculptures are variations on Torus.

Another artist is George Hart who creates geometric based sculptures.  He has created a 3 dimensional print of the famous Sierpinski triangle.  He is also known for his constructive geometric forms created from patterns and relationships.

Zachery Abel is a bit different in that he creates his works from everyday objects such as paperclips and those large clips you use with thick packets of paper.  Zachery is a lecturer in mathematics for computer science at MIT. He does hold a doctorate in mathematics. If you check out his website, you can click on pictures of his works where he explains how he made the sculpture and the mathematical idea behind it.

Check out Bathsheba Grossman, an artist who uses computer programs to crate 3 dimensional printed steel.  She uses pure math to create things like a Klein bottle opener, a gyroid, A Borromean rings Seifert Surface, etc.  These sculptures are fantastic and fascinating.

If you read the paper by Baez, you can discover so many more sculptures who have produced mathematically based sculptures since the 1950's although there are some works from the early 1910's which could be classified in this group.

I am impressed with the huge number of artists out there who produced mathematically based sculptures out of so many different materials.  I could see having students research various artists, their sculptures and contribute a page or two to a book created by the whole class on mathematical sculptures.

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

Car Production

  My curiosity in how math is used in real life leads me to explore a variety of topics.  I'm still working on a couple more in regard to art but I'm pausing to look at the math involved in car production. We know if car companies make mistakes you end up with the Ford Edsel, a car known as one of the worst vehicles in history. Others include the Ford Pinto or the AMC Pacer. 

So how do car companies go about deciding on new cars to market.  One of the first things done is to create a market analysis by analyzing data to determine what sells, where it sales, and the price it sells at.  In addition, they look at the horsepower needed, weight, fuel economy, and size before sending their detailed ideas to engineering.  This mathematically based information that dictates the finished product.

In engineering, they use computer aided modeling to build the chassis inside the computer before virtually crash testing it again and again using calculus and physics based algorithms.  Furthermore, they use geometry and trigonometry are used to design the suspension and anywhere there is a load bearing component fastened to another.

As for the exterior design, they use fluid flow programs to check on wind flow under, over, and around the vehicle.  This particular computer program allows the engineers to fine tune the aerodynamics of the car design.  Geometry and trig is used when designing the interior of the car.  Its used to help make the interior appealing to the perspective buyer and the correct placement of pillars, dashboard, dials, steering wheel, radio, etc.

Once everything is set, its time to build the cars.  In the old days, men did everything but now many cars are build by robots who are controlled by computer calculations so everything is accurate to the nano-meter so everything fits precisely.  

Once the car is built, the math is not done.  A certain number of cars are crash tested while others are taken out and road tested.  Those that are road tested have precise measurements taken of its ride, handling, emissions, etc.  These measurements are analyzed mathematically in addition to figuring out delivery schedules, delivery costs, and is sent to the dealership.

Math is used from start to finish and even beyond.  When you take your car in to be fixed, you are often charged for parts and a per hour cost for labor.  The car and its production has come a long way from the Model T build by Henry Ford.

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

Happy Pi Day

  Pi is probably the most famous ratio in history.  It is also one of the most famous irrational number known to man.  Today is the day designated to celebrate the awesomeness of pi.

As you know, pi is the ratio of circumference to diameter which means that no matter the size of the circle, the ratio is always the same.

Although the concept of pi has been around for a long time, it was only in 1706 that William Jones first used the Greek symbol for pi.  But The symbol did not come into popular use until Leonhard Euler used it in 1734.  The letter p in pi represents the perimeter of a circle.

Interesting fact:  Pi day became an officially recognized by the United States Government when House Bill 224 passed the first session of the 111th Congress of the United States in 2009.  March 14th is the perfect day for pi since its set for the 14th day of the 3rd month or 3.14.

Now for some interesting facts about pi.

1.  Pi day is also Albert Einstein's birthday along with several other famous people.

2.  If you were to print the first 1 billion digits in regular font, it would cover the distance from New York City to Kansas City.

3. "I prefer pi" is a palindrome.

4. In "Wolf in the Fold" from the original Star Trek, Spock beats the evil computer by having it calculate pi to the last digit.

5.  Pi as the secret code plays a part in two movies - The Net with Sandra Bulluck and Torn Curtain by Alfred Hitchcock.

6. There are no zeros in the first 31 digits of pi.

7. Givanchy marketed a cologne named Pi.

8. It is said that everybody's birthday appears somewhere in the digits in pi in order.

9.  At position 763, there are 6 nines in a row.  This is known as Feynman point.

10. The longest sequence of numbers to appear in order is 12345 and it appears eight separate times.

11. Pi has been calculated to the 1.24 trillion digits and  calculations continue.

Have a wonderful Pi Day.  

Remediation in Mathmatics.

  The other night, I was reading a book on working with students who are well below grade level.  The usual practice is to begin them where they are and work to get them caught up but the author of the book said this was not a good idea because they'd never get caught up.

This lead me to wonder what are some good ways to work with students who need remediation while allowing them to learn the new material just like their classmates.

Most students who spend time on remediation based computer programs know they are behind.  Many have been working on the same computer program for a couple of years, yet they are not caught up yet.  Perhaps they feel as if they may never catch up.

One article suggests increasing math rigor rather than slowing down. Teachers should intensify their instruction develop their abilities in math, develop better recall, improve learning behaviors, and help them move beyond solving problems using a memorized series of steps.  In addition, it is suggested teachers help motivate students so they move past their belief that they cannot do math.  Furthermore, instruction should include conceptual learning so a student has multiple ways to solve problems while providing opportunities for critical thinking and helping them connect to various concepts.

While looking at the topic, I came across the phrase "remediation through acceleration". Remediation is having students work on learning concepts from the past while acceleration is having students learn the material before the others in the class.  This process gives struggling students a chance to stay up with their classmates.  This is often done in a second class taught earlier than the regular math class.

Its well known students learn better when they have some prior knowledge of the concept and there is quite a lot of research supporting this.  The idea is to expose students to the new concept so they have a chance to build prior knowledge before they are taught the topic in class.  Acceleration does visit basic skills but only in the context of those skills to be immediately applied to newest concept.

Often times, the lack of prior knowledge is connected to vocabulary development.  So it is important to include vocabulary development of critical terms.  A student who has a rich understanding of a topic, when asked to write down what they know, will make a list using proper vocabulary to describe the topic.

For acceleration to succeed, the teacher needs to figure out exactly what skills and vocabulary the student needs in order to learn the new concept.  The goal of accelerated learning is to have students:

1. Understand the purpose of the concept and real world connections,
2. Acquire critical vocabulary
3. Learned the basic skills needed.
4. Learned the new skills needed.
5. An idea of where instruction is headed.

When implementing this method, it is important to identify which students should be in it, deciding who will teach the class and when.  Essentially, these students will be taking two math classes each day, one to help build the skills they need for the second class which will provide regular instruction as normal.

This is an interesting idea.  In essence, it is providing scaffolding to help students keep up with their classmates rather than separating them into a slower "remedial" class where they feel as if they are not as smart as others.  I'd love to hear what you think.  Let me know.

Artists and Mathematics

  I discovered another two or three ways art is connected to math but I'm not counting perspective, scales or anything like that.  I'm referring to art work which has a particular mathematical slant but may not be created by mathematicians.

If you ever studied art history in high school or college, you might remember the cubism movement from the early 1900's.

The two most famous artists of that movement were Pablo Picasso and George Braque who began the movement.  The name appears to have come from a comment on Braque who "reduced everything to geometric lines, cubes".

The artists broke everything down into planes so they could show different viewpoints at the same time in the same space using lines, angles, and shapes to create their distinctive style.

On the other hand, check out a more recent artist by the name of Frank Stella who created art through the use of irregular polygons that are bright and festive.  He is an American born artist who spent several years in the 1960's creating art made up of lines, circles, etc.

 His polygons not only have different length sides but they are also repeated patterns of broken circles or stripes that flow in geometric shapes.  One is a square divided into four isosceles triangles  using two diagonals.  In each quadrant, there are stripes of two alternating colors going to the center.

Another one is horizontal stripes broken by rhombus divided into four triangles with the lines going outwards in an x shape so the lines meet the horizontal lines.  Its in black and white and really really cool.  Within that three year period, he created some wonderful pictures that used only geometric shapes and are awesome.

Other artists to look at are Simon Beck who creates art like Koch snowflake or Sierpinski triangle on snow using nothing more than a compass and his snow shoes. This art is large and covers a huge area.  Its like he translates a small picture into a larger model.  His art is fantastic and quite realistic. Then there is Hamid Naderi Yeganeh who uses computer programs based on mathematical formulas to produce computer generated art work which is intricate and three dimensional in appearance.

Check out Tom Beddard who creates Faberge Fractals.  He generates them by using the output from one time as the input for the next run in a iterative formula.  The art is quite detailed and absolutely breathtaking.  Did you know there are different types of fractals?  Each fractal produces a different type of picture.  For instance, the L-systems produce a fern looking plant.  Check this site out for more information on this.

Think about sharing these artists and their art with students to show them how mathematics can produce beautiful work worthy of being shown in galleries.  I think its important to show students more than just the mathematics themselves.  Sometimes you have to venture outside the box to give students an appreciation of the whole topic. 

Its Done

Springing Forward.

Two Highly Effective Learning Strategies.

  Its interesting what researchers say about 5  effective learning techniques of which two are considered highly effective. 

The first highly effective technique is practice testing where students are frequently tested with low stakes quizzes.  Practice testing is another name for retrieval practice and should be carried out over a period of time because it helps material stick. 

The important part is to make sure theses tests or quizzes are worth few to no points because it allows students to practice in a safe environment.  This type of frequent retrieval practice improves long term ability to remember the information.  The best type of test to use is one that allows students to freely recall the information.

Research indicates if students are tested right after reading a passage remembered more than those who simply reread the passage.  In addition to remembering more, students were able to apply the information in new ways.

Frequent quizzes can be applied to any subject and any age group. In addition, teachers should provide a way for students to retest over days or months rather than doing it all on the same day.  It is also important to provide feedback on student performance once the test is completed rather than after each test item.

Other ways to implement practice testing are to have students write down the main points of the day's lesson during the last few minutes of class.  Teachers should also pretest students before introducing the material because it prepares a student's brain for the new material.

The other highly effective technique is distributed practice or spacing.  This practice has the teacher spreading out the concept over a period of time rather than just at once.  By spacing the study over time, students can increase their information recall by up to ten percent.  Again, this technique can be used for all ages in all subjects.

The way an educator implements this is by introducing a new topic one day and then return to the topic over time.  Furthermore, they should take place after the material has been taught so students are forced to retrieve it from their brain.  This technique is more effective than just doing a bunch of problems or study all at once.  This doing problems all at once is referred to as massed practice.

The idea is that each time you come across the material during study, your brain tries to retrieve the material.  If the retrieval is successful, the memory is more likely to remain rather than being forgotten.  Another theory on why distributed practice works is that when you learn the material, your brain encodes it with contextual information such as how you are feeling, how the information is presented etc.  Because it is being done over time, the contexts are more variable and provides better cues for retrieval than massed practice.

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

Making Sense of Area Formulas.

  The other day in Geometry, one of my students asked how they got the area formula for a kite and a rhombus because essentially the formulas are the same, yet the shapes are slightly different.  Another student asked about the area formula of a trapezoid and where it came from.

So out came the paper and the scissors to cut out shapes and how how they related.  The kids watched every step and I saw lots of 'Aha' moments sparkle across the room as light bulb after light bulb exploded.

For both the kite and the rhombus, I drew a rhombus inside a regular square, cut the rhombus out before flipping the cut pieces around to cover the rhombus.

The students saw that the diagonals were actually the height and base or length and width of a square.  They saw then  the cut pieces made a second rhombus and that is where the divide by 2 or multiply by 1/2 came from.

When I repeated it with a kite inside a rectangle, they were like wow.  They saw how the cut pieces when place over the kite, produced a second kite of exactly the same size.  They also saw the diagonals on the kite also represented the height and base of a rectangle and the rectangle produced two kites thus the division by two in the formula.

As for the trapezoid, I placed two right next to each other, one right side up, one flipped so the two of them together formed a rectangle. 

I labeled various parts as base 1 and base 2 along with height.  One young man, declared you found the area by adding the length of the bases together to form one side multiplying that number by the height.  Since the trapezoid was half of the rectangle, you had to divide the answer by two. 

I think next year, I'm going to create a hands on activity where students get to explore area formula's so they derive them by themselves.  This will help them gain a better understanding of the how the formulas come to be.

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

Asking Questions More Effectively.

Sometimes, its hard to use effective questioning on students when you didn't get it during your teacher training program.  I am having to learn to ask effective questions but I still stumble at it.

There are eight things you can do to prepare yourself to ask more effective questions but they aren't all that easy.

First, as you write the lesson give some thought to how your students will interpret the new material.  Try to anticipate their thinking and all the ways they might solve the problem.  This can help you plan on the questions you might ask so you encourage deeper understanding.

Second, connect your instruction to the learning goals formed by the curriculum.  These goals help the teacher determine the questions to be asked and the problems used to reinforce learning goals. By asking questions which go back to the curriculum, students are able to focus on key ideas and principles. 

Third, include more open ended questions because they challenge student learning and supports learning.  These type of questions encourage a multitude of strategies and responses.  An open ended question might be 'How many different ways can you draw an area of 48 square units?' vs closed which might be 'How many sides does a quadrilateral have?'

Fourth, make sure the questions you ask can be answered and do not provide students with the answer.  They need to think about the question to find the answer.

Fifth, use verbs such as observe, connect, justify and others which demand higher levels of thinking.  These verbs need to encourage students to share their thinking,  while deepening their understanding and extend their learning.

Sixth, ask questions in such a way as to open up conversation among all the students and not just between the teacher and students.  These questions should have students discuss their thinking behind their solutions so as to build on prior knowledge and relates to new learning.

Seventh, make sure questions are neutral and do not contain words such as hard or easy which might shut a student down before they try.  Teachers should also monitor their facial expression so they are not giving out nonverbal clues which could shut a student down.

Finally, As stated earlier this week, allow enough wait time for students to pull their thoughts together to answer.  Students, especially English Language Learners may need extra time to pull together their thinking.  Do not be afraid of silence.

I think all good teachers want to improve their teaching style but sometimes it becomes difficult and we revert to the way we experienced questioning when we were in school.  Let me know what you think, I'd love to hear.

Demanding the How or Why.

With the advent of apps which solve problems for students, what can we do to create deeper understanding and a knowledge of why problems are solved in a certain order.  I've had a student who used an online calculator to factor trinomials. 

It was easy to tell she'd done that.  She didn't show her work and she bombed a quiz I gave her.

I'm starting to ask questions which not 'What is the next step?' but 'How do you know this?"  I'm asking for evidence of them stopping to really think about what they are doing.

My Algebra I students are currently working on solving systems linear inequalities.  I start by asking what type of boundary they will be using in each problem.  Then I ask them to tell me how they know they are right.  This is where they might say its a 'dashed line because the inequality does not have an equal sign'.  The ones who can answer this question are doing much better than those who want to just do it without thinking.

According to an article I read, we do not want to ask simple information type questions such as what is the formula for area.  That is simple recall and can be looked it up in the book.  The author says these questions can often be used to see what the student knows but they do not require higher order thinking.

It is better to ask questions which require students to elaborate, explain or show their thinking.  These types of questions require the student to explain the steps they took to solve a problem or their thinking behind the method they chose to solve it.  Furthermore, give students sufficient time to put together their thoughts so they can explain. Unfortunately, one average, students are only given five seconds to answer.

Require students to connect mathematics to relationships as they answer such as the visual representation of the coordinate plane of an inequality with the actual equation.  Encourage them to reflect on their thinking and justify their choices.  Have them explain why they chose a certain method to solve the problem.

Do not steer the conversation to a desired outcome because the student may not have chosen that way of doing it.  When we funnel the conversation, we ask very specific questions to steer the conversation but save the higher level questions for later and do fewer.  Use questions which blend questioning, reflection, justification and probing.  The questions have students share what they notice while encouraging them to share their thoughts.

Teach students to ask each other more open ended questions such as 'Why do you think that?' or 'Could you have solved it a different way?'.  This technique helps students begin discussing mathematics which helps create a deeper understanding. 

I'd love to hear what you think.  Have a great day.

The Fibonacci Sequence and Nature.

We all have those students.  You know, the ones who spend the period recreating the latest Manga figure, or want to recreate a picture they saw on the internet.  It doesn't matter what class they are in, they'd rather draw.

I have some who are extremely talented at recreating artwork but have no real interest in the regular academic subjects.  So how do we interest them in math when they only see the theoretical equations and not applications.

In fact, we read that nature is a real life application of the Fibonacci sequence but how does it get from a formula to the flowers.  If you remember, the Fibonacci sequence begins with 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, etc.  I understand the sequence mathematically but I do not understand how it translates to nature.

According to a Science article, sunflowers show certain parts of the Fibonacci sequence beautifully for either 34 and 55, or 55 and 89, or 89 and 144 depending on the size of the sunflowers.  These numbers represent the spirals of seeds from the center to the outer edge.  It doesn't matter if you go clockwise or counterclockwise, those are the two numbers which repeat.

For the past four years, the Museum of Science and Industry in Manchester, England has had members of the public growing sunflowers, taking pictures, counting seed spirals, and reporting back their results. After checking the results of over 650 flowers, scientists have discovered that one in five flowers produced either a non-Fibonacci sequence or one that is more complex than the usual.

Now for the actual mathematics of sunflower heads and other flowers.  The spiral is produced by a slight fractional turn. For instance if you choose a 90 degree turn it is 1/4th of a circle but in reality the seeds are placed in irrational fractions such as 2/3, 3/5, 5/8, 8/13, 13/21, etc.  Notice how the digits in these fractions relate to the Fibonacci sequence.  A relationship of how the Fibonacci sequence relates to the creations of sunflower spirals and other spirals in nature.

According to another article I read, the number of petals a flower contains is one from the Fibonacci sequence or if the flower is more complex, you can count spiraling petals either counterclockwise or clockwise and still discover numbers from the sequence.  In addition, certain flowers have a specific number of petals. This article shows how to count the spiraling pedals and has several pictures for students to practice on.

This article explains more about the number of petals and their placement within the flowers. For instance, the 8 petal rose has a center, three petals around that and five which surround that.  Each set of added petals are added in a new level.  As far as real flowers go, the marigold has 13 petals which the daisy has 21, 34, 55, or 89 petals arranged in levels.

This phenomenon is not just seen in flowers.  It can be seen in pine cones, the vegetable Romanesco, pineapples and so many more things.

Now how do you involve your artists?  Let them draw flowers any way they want and then compare their drawings with the real life photos to see if their's looks as "right" as those in the photos.  Introduce the Fibonacci Sequence to them mathematically, then discuss it with pictures and activities letting them see how prevalent it is in nature.  Let them redraw their flowers so they are done with the correct sequences to look more  correct.

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

Warm-up

Warm-up

No Kill and Drill

  Unfortunately, there is a segment of the population who still believe the old ways are the best.  Lecture on something new every day and have them practice tons of problems within a short time before moving on.

Unfortunately, this does not mean that students will truly learn the material.  I discovered the other night that some people feel we should not integrate technology into the classroom to the extent of making students more independent because these same students could easily find the answers on the internet.

Yes, they could but they don't know what they are doing if you throw in a quiz here and there or if you require them to explain each and every steps and require justification for each step.  In addition, it is felt that if I as a teacher are not requiring 20 to 30 problems per day, I am not doing my job.

Current research indicates that we should not be assigning tons of the same type of problems to the students each day.  What we should be doing is assigning several problems arranged in such a way that no two problems in a row are solved in the same manner. For instance, you might have them find the area of a circle, solving a multi-step equation, finding the answer to 6^3, graphing a linear equation, identifying the transformation then start again.

Then this is done, it forces students to choose a strategy based on the problem itself rather than knowing that is the strategy being used in the section.  It creates a more real life situation such as you might find in a job.  It allows them to learn what they need to know.

Most schools have textbooks are still set up to introduce new material complete with lots of problems that all use that strategy to solve.  Because the problems are not mixed students do not get to practice selecting a strategy and the solution to that problem is found by choosing the correct strategy.  In addition, many problems are similar but require different strategies to solve.  Understanding this, helps students read problems more carefully.

With most textbooks, students know they have to use a certain strategy to solve all the problems in that group so they do not have to read the problem.  Furthermore, this type of practice means students believe they have learned to solve the problems when in fact they do not because they have not had the opportunity to select a strategy.  Even in mixed review sections, the publishers have grouped several problems of the same type together.  In other words, students are trained to just do the problems all the same way so when they get a test where the problem types are mixed up, they do not do well.

One way around this is to make a copy the mixed review problems or several problems from different sections of the text book, cut them into individual problems, tape them on a new paper and make copies, if you are still doing things the old fashioned way.  If not, take the problems and create a quiz with no two problems requiring the same strategy next to each other.

Another way is to just assign a bunch of different problems from different pages such as page 32 #5, page 45 # 10, page 61 # 12, etc. No matter which way you choose to do it, you are helping students to really look at the problems to choose a strategy rather than just blindly doing problem after problem as is normal.

So should all their practice be done this way?  No, it is recommended that one third of the problems be interleaved but the first few problems should all be the same so they get practice in the strategy.  When giving interleaved problems, students should be given immediate feedback so they can make corrections and ask questions for clarification.

Interleaved practice does raise scores on tests which contain a bunch of different problems much in the same way as a cumulative test is set up.  It helps students learn to both choose and use a strategy which is something they need for most high stake tests.

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

Number Systems Other Than Base 10.

  Although most countries use base ten number systems, there are several civilizations and situations which do not use the same base.

The Oksapmin people of Papua-New Guinea use a base 27 number system because they count 27 different body parts from the thumb to nose, the wrist, etc.

The people in Mexico who speak Tzotzil use a base 20 counting system by counting all their fingers and toes.  Once they get to 21their language describes it as the first digit of the next man, 22 is the first two digits of the next man, etc. 

The African language, Yoruba, has a base 20 number system but they add the digits 1 to 4 to the 10, 20, etc so 13 is 10 plus 3 but then it subtracts for 5 to 9 so 15 might be 20 minus 5 but 78 might be 20 x 4 minus 2.

Another Papua-New Guinea language, Alamblak uses only the numbers 1, 2, 5, and 20 to create all the other numbers so thre is 1 + 2, 27 is 20 + 5 + 2 + 2 while 14 is 2 x 5 + 2 + 2.

If you speak Bukiyip, another Papua-New Guinea language, you'll use either base 3 or base 4 depending on what you are counting. Things like coconuts, and fish are counted in base 3 while Betel nuts and bananas are counted in Base 4.

Papua-New Guinea has other languages such as Ndom who has a base 6 number system or Huli which is base 15.

The Babylonians are known for their base 60 while the Mayans used a base 20 system. Even the modern French language shows remnants of a base 20 because 80 is 20 times 4, 81 is 20 times 4 plus 1 etc.

Its interesting to discover that although our society is base 10 with its decimal money system, the metric world, our counting system, there are still places out there whose language does not have a base 10 system.  I find that quite interesting.

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

If you do much with computers, you know they use a binary system (base 2) which represents true or false, on or off, and a hexadecimal based (base 16) system to express colors on web pages.