New Years Eve

 Happy New Year's Eve.

Resolutions For Math

It's that time of the year again, when most people make their New Years resolutions and about 80 percent of those same people give up by February.  I got to thinking that maybe we can help students learn to use the same criteria for successful goal setting.  I discovered early on that my students can make goals but they haven't learned to figure out how to meet the goal.  For instance, they might say, they want an A in class but they don't know how to do it.

I came across the SMART system for writing resolutions which can also be used to set realistic goals and I think it is something we can introduce our students to so they become more successful.  SMART stands for specific, measurable, attainable, relevant, and timely.  

When we say specific, it means to set the goal and be specific about the goal.  Setting the goal of getting an A is not specific but stating they will complete half of all the assigned problems is specific, especially if they seldom turn in any work.  Let them know the goal is what they hope to accomplish and they need to address what challenges they will face, and where they expect to see themselves at the end of the semester or year.  They need to express what the completed goal will look like at the end.

Measurable means stating what criteria is being used to "see" how the person is making progress towards reading the goal.  How will they monitor their progress? If they decided on working towards a C when they struggle with getting a D normally, they can use daily work, quizzes, tests, and any projects to monitor their progress.  

Attainable refers to setting the goal at a level the person can reach and writing down the steps needed to make that goal.  If the goal is to make 70 percent on all assignments which is attainable, the student might then determine the steps are to do all the problems for each assignment, ask for help as needed, be willing to come in after school, and maybe even redo all missed problems.  On the other hand, if a student seldom works, setting a goal for an A may not be attainable so it all depends on the student.

The goal needs to be relevant to the student rather than the teacher.  The student needs to be able to express why the goal is important and what impact it will have on a students life.  The impact might be that it helps them get into collage or it might be the first time the student got a C in a math class.  The reasons have to be student centered and should not be set because their friends are choosing it.  They need to look at their strengths and weakness when they set the goal.  It might be something as simple as memorizing their multiplication table if they don't know it or it might be learning to solve two step equations.

Finally is timely which refers to  setting a time limit and in school, that is set by the length of the semester.   The goal needs to have an ending time so the student might say they want to know their multiplication table by the end of the first semester so they know when the deadline is. Students should also be aware that the time line might consist of a series of deadlines such as knowing all the multiplication facts for ones the first week,  ones and twos by the end of the second week, adding one set each week so by the end of the quarter, they know everything from 1 x 1 to 9 x 9. 

This site offers a view of what a graphic organizer for SMART might look like and it says it offers a free version.  If it is written down, students can revisit them at the end of the year and see if they made their goals.  Let me know what you think, I'd love to hear.  Have a great day. 

New Years Math Fun Facts

 

Today, I decided to look at fun math facts associated with New Years because they can be used to help hook students when we get back to school for spring semester.  For most people, they stay up till midnight, enjoy a glass fo something to ring in the new year.  In addition, you might watch the various new year celebrations that are broadcast beginning on the east coast and follows across the country so you can celebrate more than once.

Have you ever wondered how many times the new year is celebrated around the globe?  When they set up time zones, they wanted to set it up as 24 different zones, each one hour apart but it didn't work out that way.  Instead, many of the countries are half an hour off of other countries, or a country has multiple time zones so it works out that there are 40 different countdowns to ring in the new year.  But what happens if you are on the International Space Station?  Since they are not associated with an earth based time zone, when do they celebrate the new year?

Since the International Space Station is moving all the time, they move fast enough to see sunrise and sunset every 90 minutes which mean they spend about 4 minutes in each time zone. This can make it difficult so they base their time and New Years celebration on Greenwich mean time because the Royal Observatory has been at the center of time keeping for centuries.

Did you know that 45 percent of Americans make resolutions such as loosing weight, giving up smoking, save more, etc but 25 percent of those give up on their resolutions by the second week of January.  So if the population of the United States in 2020 is said to be 329.5 million people, how many people made resolutions and how many quit within two weeks.  According to another fact, 80 percent of people give up on their resolutions by February. 

In addition, Americans open enough champagne to enjoy 360 million glasses of it at the New Years.  If we assume only adults drink the champagne, we know that 258.3 million people of the United States are adults so how many glasses are being drunk per person?  Furthermore, 41 percent of Americans state that New Years eve is their favorite holiday but 3 percent of the population never celebrates it at all.  In addition, 28 percent of people who drink, have food delivered on January first.

We all love watching the ball drop in New York city.  The one they televise every New Years at midnight and all one million people in Times Square celebrate it.  Times Square is said to take up 11,600 square feet  of area so the density of squeezing one million people into it is going to be huge.  Let your students calculate the density of number of people per square feet.  Since it works out to about 86.2 people per square foot, it can open up discussions on how can you get that number of people in such a small space.  This million people leave 56 tons of trash including 1.5 tons of confetti.  This opens up another great problem of figuring out how many pounds of trash are left by each person?

These facts offer teachers a chance to turn percents into actual numbers so students get a more realistic idea of how many are participating in each thing.  Let me know what you think, I'd love to hear.  Have a great day.

The Day After

 The day after

Merry Christmas

 

Happy Christmas Eve

 

Don’t forget to put out the cookies.

Ancient Greek Astronomical Calculator

 

Just imagine if you will, the ancient Greeks were able to create what is considered the world's first analog computer. This happened around 2,000 years ago when they created the Antikythera Mechanism which was a hand powered device used to predict astronomical events.

What is quite fascinating about the device is that it is considered one of the most complex pieces of engineering to survive from so long ago. The ancient Greeks used it to predict the position of the sun, moon, planets, and the lunar and solar eclipses.

This device was found in 1901 by sponge divers in a shipwreck dating back to Roman times, in a location near the island of Antikythera. They found a device made out of bronze with 30 bronze gears used in  complex patterns  to determine astronomical events.

Scientists have studied this device since it was discovered but it wasn't until 2005 they were able to determine how it worked after using three  dimensional x-rays combined with surface imaging to create a better picture of how the mechanisms actually predicted eclipses and how it calculated the motion of the moon.  However, it wasn't until earlier this year that scientists were able to understand how the full gearing system in the front worked because only one third of the mechanism survived but it was split into 82 pieces.

Some of the results from the research done in 2005, revealed thousands of text characters hidden inside the fragments or pieces and the text has not been seen in over 2000 years. They found a description the back cover which had a description of the heavenly display complete with planets moving along rings represented by marker beads.  In addition, the largest fragment, Fragment A, showed the features of the bearings, pillars, and blocks while another large fragment, Fragment D, showed a disk, a 63 tooth gear and plate. 

A group of people from UCL in London relied on the description on the back cover to recreate the display.  They also relied on the numbers 462 and 442 from the front cover which seem to represent the planetary cycles for Venus and Saturn and these numbers were quite accurate especially since the classic astronomy of the first 1000 years BC originated in Babylon and there is a question as to how the ancient Greeks, arrived at the accurate values for Venus and Saturn.

So the team used an ancient mathematical method used by the Greeks to derive the length of cycles of Venus and Saturn, and also managed to determine the cycles of all the other planets.  The method was described by philosopher Parmendies with enough detail, the team could use it. Furthermore, they took the evidence from Fragment A and Fragment D to the 462 year cycle for Venus and the 63 tooth gear and plate played an important roll in this.

This lead the team to creating a mechanism for each planet that could be used to calculate new advanced astronomical cycles while minimizing the number of gears in the whole system. This has helped science move one step further to fully understanding the device.  Isn't this impressive?  A device that accurately calculated astronomical entities?  Let me know what you think, I'd love to hear.  Have a great day.

Teaching About Hearts

This topic is a great one to introduce around valentines day since there are multiple equations one can use to generate different hearts.  I realize hearts is not the proper mathematical term but it is one that students can relate to and once the topic is introduced, we can sneak in the proper term of cardioid. 

A cardioid is defined as a fixed point on the circumference of one circle with radius r that is rolling around the circumference of a second circle with the same radius and does not slip thus forming something similar to the cross section of an apple. 

For some math classes we teach, we don't need to get that into the explanations but I found a site from Denmark that explores the heart curve in such a way, we can introduce students to the topic in a way they understand. The site defines a heart curve as a closed curve which is the shape of a heart such as the hearts found on a deck of cards. Furthermore, many of the directions they give, combine geometric shapes to produce the final shape.

In the first part of the site, they show how to easily draw three different types of hearts including one that is closer to the human heart and is a good way to integrate art into the math classroom.  The easiest is using a square that stands on a point with a half circle whose diameter is the same length as the side of the square on two sides.  Then they give five different instructions for drawn hearts including two that use restricted sin waves, reflection, rotation, and semicircles. These use solid geometric vocabulary. 

The next section is where they talk about calculated heart curves where they list how to use certain formulas and restricted domains to create a dozen or so hearts and at least one three dimensional heart. They show the equation and the heart the equation produces so students can use a program such as demos to play around with the equations and experiment.  Furthermore, there is information on the Mandelbrot set and cardioid or weaving things to produce heart shaped final products. Finally, it offers some real life places one can find hearts and how they were created such as in window grills in Venice, or the heart shaped rosettes in Venice.  

If on the other hand, you want to provide the equations so students can graph them to see what happens, head for Wolfram because they have a lovely page with seven different equations with drawings of the final product and more detail on the actual equations.  It is more of a reference page but it provides teachers with what they need to create a lesson or two on heart shapes.

The New York Times ran a nice article on the Perfect Valentine back in 2019 and in the article, they have a link to a site where students can change the sliders to change the heart or they can change parts of the equation to see what happens when you change squares to cubes or changes the constants to see how it all changes the beginning heart. The best thing about the widget is that the creation is three dimensional.  

Valentines and Cardioids are related and such a great place to throw in some art work, serious mathematics and fun.  Let me know what you think, I'd love to hear.  Have a great day.

Warm up

 

A doughnut is a type of torus. Look up the formula for the volume of a torus. Explain what the variables mean.

Warm-up

 

What is the mathematical name for a heart?

Math Jokes

 

Everyone loves a good joke.  I try to integrate math based jokes in class to show my students that math teachers are not always dour and stern.  Today, I'm sharing some that I use in class with my students.  Sometimes students laugh but other times they look at me with the same amount off disbelief they display when they discover I know about tiktok, instagram, or snap chat. 

1. Why do plants hate math?  Because it gives them square roots.

2. Why did the triangle make the basketball team?  It always made 3-pointers.

3. Have you ever noticed what's odd?  Every other whole number.

4. A student turned in a blank sheet of paper for his math test, and the teacher asked him why?   "It was on imaginary numbers" he said, "Can't you seen them?

5. Why can't the angle get a loan?  It's parents wouldn't cosine.

6. What do you call a mathematician who spent all summer at the beach?  A tan gent.

7. What do you call the political party in favor of agriculture? Pro-tractors.

8. There's a fine line between a numerator and a denominator. But only a fraction would understand.

9. What do a year and a dollar have in common?  They both have four quarters.

10. After a sheepdog chased all of the sheep into the pen, he told the farmer.  "All 40 accounted for."   "But I only have 36 sheep" the farmer replied.  "I know" said the sheepdog. "I rounded up."

11.  Why did two 4's skip lunch?  They already 8.

12. How is an artificial Christmas tree like the fourth root of -68?  Neither has real roots.

13. Teacher:  "Why are you doing your multiplication on the floor?"  Student "You told me not to use tables."

14. Why do mathematicians like parks?  Because of the natural logs.

15. Why should the number 288 never be mentioned? Because its two gross.

These are just a few jokes to get you started if you want to entertain your students.  If you do a search, you'll find quite a few on the web if you need more.  Let me know what you think, I'd love to hear.  Have a great day.

The Numbers Never Lie.

The numbers never lie is a line from a film I managed to watch on my flight from Seattle to Dulles Airport in Washington D.C.  Picture two guys dressed in homemade space suits sitting in a car, with the windows rolled up.  There are three rocket engines attached to the two sides and top.  The car is sitting on the same type of huge airplane that took the space shuttle up.  One of the guys is concerned it won't work and the other guy holds up his cell phone stating "The numbers never lie".  In reality, this situation would never happen since it is physically impossible.

In reality, numbers can lie to us or should I say, the presentation of numbers can lead us to believe things that are wrong.  This becomes especially important when looking at articles that present certain "facts". I watched a program one time where the doctor stated that coconut oil is good for us but I did some searching and discovered there is only one study out there supporting this statement and it only looked at 30 people.  This study had such a small sample size that the conclusion was not valid and that is one of the ways numbers can lie.  

A small sample size might be the number of people studied but it might also be sports writers coming to a conclusion about a players ability after a couple of weeks of play.  So it is always a good idea to see how many people were involved in the study, or how long a period of time the data is for when looking at a sports person. 

Another way numbers lie is when large meaningless numbers are thrown around. A good example of this is when looking at the number of followers an "influencer" has. The number may not reflect the actual number because there are many ways to increase the numbers of followers without caring if they all care.  I've personally seen where social myth busters have gone through and shown how one influencer increased their number of followers with entities who may have been created by her mother.

The third situation in which numbers may be lying involve correlation rather than causation. This is when one thing is said to cause something else when they are actually just a correlation.  An example would be something like saying that a large number of male students wake up with headaches, are still wearing their shoes so you wonder if the headaches are caused by wearing the shoes but we know that isn't true.  It is a correlation, not a causation because wearing shoes do not cause headaches. 

The next situation is where the numbers used imply the data comes from a random sample but in reality it came from a non-random sample. An example of this, is when the data comes from an online voting site. The data from one of these are not random because the people who participate tend to read the website that is hosting the survey and that means only certain members of the whole population are involved, not a good cross section of the population. 

Then there is adjusting the visual so it "looks" as if the data is supporting the hypothesis.  This happens when they change the values of the y-axis so the differences look bigger or smaller than they actually are.  The last way is to choose the lowest and highest values to be used in the analysis of the data.  We often see this type of adjusting with the top 10 lists, or the top 5 ways, etc.  It is not important for many things but if you do it for say the top 10 colleges, the information used to determine this list may create issues for readers if not done correctly.

Contrary to the character's line, numbers can and do lie based on it's presentation to people. Let me know what you think, I'd love to hear.  Have a great day.

Using Games To Help Students Improve Math Skills

 

We all know that students love playing games, especially video games.  I came into work in the morning and students are all on their phones playing a variety of games.  Many adults also enjoy playing games but the type of game depends on their age and interests.  As teachers we know we should be including games in the classroom to make it more interesting and help keep student attention. We can use both video games and personal interaction games such as Uno, or a board game.

Math games are great for helping students improve their math skills because they are like most other games in that they have set rules, goals, and a level of competition.  A good math game should have just the right amount of challenge and competition so they feel as if they have a chance to win even if they lose.  A good game also lets them practice their problem solving skills while making decisions and are easy to learn.  In addition, students feel less pressured to learn math when they play games. The games used in math might be digitally based, boardgames, physical movement games, or a mixture of all three.

There are four good reasons for incorporating math games of all sorts in your classroom.  One of the first reasons to use games is that it can create a mind - body connection, especially if the game involves movement.  When students move, it helps the brain activate the neurons that help with processing and retention of the material.  In addition, when students learn the material while moving, they tend to remember it longer and the fun experienced while moving helps them feel more successful.  

Second, many of our students have fallen behind due to the pandemic, and many were already behind.  If the proper game is chosen, it can help students work on catching up.  Fortunately, it is relatively easy to find software which is designed to help individual students with scaffolding so they can catch up and narrow any learning gaps.  

Third, in order to use games effectively, there has to be training available to help teachers effectively integrate games into the classroom.  Teachers have to be comfortable when using games in the classroom so students have the support they need.  

Finally, integrating games into the class make it fun and students tend to develop a love of math when its fun. Furthermore, students often forget it's math because of the fun they are having and they do not struggle, they are willing to try.

The problem is that you cannot use just any game, especially for math games.  There are sites out there that offer "Math" games but as far as I can tell, they do not address specific skills, they are just games. Some places offer "games" that turn out to be regular practice with feedback that is only yes or no but nothing to explain to students how to solve it.  

It is important to take time to make sure the game you have chosen to use in class focuses on the skill they need to practice, everyone can participate in the game, and if it is internet based, will your students all be able to play.  In addition, think about using games that have students work in teams rather than using games for individuals.  Students need the opportunity to collaborate and help each other.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If there are 3 3/4 cups of hazelnuts in each pound, how many cups will you have if you have 22 pounds of hazelnuts?

Warm-up

 

If each jar of Nutella contains 50 hazelnuts, how many hazelnuts would be used in each case of 12 jars?

From The Ashes of Buchenwald Came The First Hand Cranked Mechanical Calculator.

Picture from https://history-computer.com 
page on Curta

When we think about calculators, we think about the hand held ones that came out in the 1970's or we might vaguely remember those adding machines with the rolls of tape. We don't think of anything that might have come out of World War II.  I came across an article a couple ago on this exact topic.  

Picture if you will, a small pepper mill sized hand held calculator in the hand of a person in a black and white photo. First released in 1948, the Curta hand cranked mechanical calculator was the choice of people for the next two decades until the electronic versions hit the market.

The first calculating machines were big and heavy and as such were really desktop versions.  In 1902, Curt Herzstark was born into a family whose business was to produce and sell desktop calculator and other office equipment.  By the 1930's Curt ran the family business and travelled across Austria, Hungary, and Czechoslavakia selling machines to banks and factories. If you've seen Hidden Figures, you'll have seen one of these machines used by some of the women.  One thing he heard from many of the people such as architects, customs men, and others who wanted something that was more portable.  

After hearing all of his clients comments, he knew something needed to be done.  The calculators would need to be redesigned because the parts could not just be reduced in size and still have the machines work. He began by picturing how the new handheld calculators would look so that it could be used.  He decided it had to be a cylinder, and there had to be numbers on sliders that one could input values, one digit as a time.

So by 1937, he had worked out the design, determined how the parts should go together to work and was ready to create a prototype but the storm cloud blowing across Europe came hit his country and Hitler took over. Although his father was Jewish and his mother catholic, he was concerned but the Nazi's allows the factory to continue as long as they produced tools and machines for Germany.  This was fine until charges of supporting Jews and having an affair with an Arayan woman were lodged against him.

This resulted in his arrest and being sent to Buchenwald concentration camp where conditions were brutal.  Curt was assigned to work in a factory producing V8 rocket parts.  One of his supervising engineers heard rumors of his hand held mechanical calculator. Curt was told that he could work on his machine and when the war ended in victory.

This one thing gave him hope of surviving the conditions and he was able to draw actual plans for his machine before the Russians liberated Buchenwald in April 1945.  Curt walked to the next town over and took over an abandoned factory where he created a prototype of his machine from the plans he'd carried folded up in his pocket.

When the Russians overran the village, he escaped to Vienna with both the prototype and his plans.  In post war Europe, countries were struggling to begin again.  Luckily for him, the government of Liechtenstein found his machine interesting and they set up a company for him.  He used this company to produce his Curta hand held mechanical calculator for sale in 1948. It could perform addition, subtraction, multiplication, and division, find roots and powers.   The machine was well received and loved by all who used it because it provided the precision and mobility they wanted. He passed away in 1988.

It is said that around 150,000 machines were sold between 1948 and 1972 when small hand held electronic calculators took over the market.  It is still possible to find second hand Curta's being sold but they can be quite expensive. The other place to find these machines are in museums across the world. The world is lucky he survived the horrors of Buchenwald so he could make his mark on the world.  Let me know what you think, I'd loved to hear.

Using Math To Improve Your Pool Playing

 

I learned to play pool when my family stopped at a KOA camping ground in New Mexico.  I was a teenager, 17 or so, and it was a great way to meet boys but I wasn't real good at it.  Now when my students ask about when they'll use geometry, I tell them that you use it when you play pool because so much of it involves angles. 

From a mathematical point of view, playing pool involves a dynamic system. Mathematically, a dynamic system is a system in which a function describes the time part of a point in geometric space.

So lets look at the basics.  A straight shot is one that goes straight into the pocket and is considered one of the easiest to do. Then there is the angle shot where the player bounces the ball off the side using the law of reflection which states that the ball bounces off the side at the same angle it hits it. So if the ball hits the side at a 35 degree angle, it will bounce off at 35 degrees so knowing this you can predict where the ball will go based on the angle it approaches the rail.

Lets start with the break.  Mathematically, if you are using an 8 ball set up, you want to hit it straight on but if its a 9 ball set up, you should have your ball approach it at an angle so you are able to bump one ball into a pocket without scratching. In addition, it allows you to choose whether you want solids or stripes and it can put your opposition at a serious disadvantage. 

In addition, it is important to keep track of how much energy you use when you hit a ball with your cue.  If you hit the ball too hard, it may bounce off another ball or end up in a pocket so you are out.  If you don't use enough power, your ball won't go very far.  Furthermore, the angle you strike the ball determines where it goes. Now if you keep your cue more towards the middle of the table, it makes it easier to shoot the ball more accurately without using too much energy.

Now back to the law of reflection.  This can be used when you have the cue ball and the ball you want to hit and they are the same distance from the rail.  When you shoot the cue ball, you need to aim about half way between the two balls so the angle the cue ball hits is the same as when it bounces off.  Basically you are relying on congruent triangles.  If the balls are not the same distance from the rails, you have to rely on similar triangles to hit the object ball because the share the same angle but they travel different distances.  

A good player who understands the math also knows that that how they hit the object ball with the cue ball can determine which way both balls travel.  If the cue ball hits the object ball at the wrong angle, it could send the object ball the wrong direction while causing the cue ball to scratch.  So many different things to keep track of but its all mathematically based.  One person recommends that the easiest way to win is to play against someone who doesn't know the math involved in playing pool.  Let me know what you think I'd love to hear.  Have a great day. 

80th Anniversary Of The Bombing Of Pearl Harbor.

 

Please remember those who lost their lives 80 years ago on December 7, 1941 when Japan bombed Pearl Harbor. 

Individualizing Instruction With Technology

 

This is the second full year, schools have had to contend with COVID. The newest mutation has just landed in the United States and many school districts and schools are still flipping between virtual, hybrid, and in person instructions.  We are aware students have holes in their foundational knowledge so it is important to think of ways to help students catch up from where they are, not continue pushing forward without a chance to fill in those holes.  

One should consider a computer based program that utilizes computer algorithms and machine learning to offer students daily individualized instruction. Such programs are also known as adaptive learning.  This type of program truly offers personalized learning without having to sit down and prepare plans containing material for every single student.

The way this type of program works is that students log on to take a test to identify what they know, what skills they have, areas where there are gaps, and uses the data from the test to create daily lessons.  Some of the programs even determine how the material should be presented while others use a standardized method of presentation.  In addition to identifying gaps in knowledge, such programs can also help students who are on or above grade level to continue forward without stagnating. 

Furthermore, most of these adaptive learning programs require that students master the skills before moving on and they offer different pathways for the student to accomplish this and there is usually a way for the teacher to monitor a students progress while keeping up with newly identified gaps, etc because the adaptive technology is continually monitoring.

This type of program is perfect for this pandemic because we've reached a point where most schools have something set up to make sure all students have access to the internet at home should the class suddenly go virtual.  Furthermore, it identifies a starting point for the student when they first log into the program and provides instruction, differentiation, scaffolding, reinforcement, and practice so students can move forward.

The important thing to remember is that these types of programs are not designed to be used as the only way to teach students.  They are designed to be used in tandem with whole group and small group instruction.  I have been told that I am teaching a combined 7th and 8th grade math class beginning in January.  I have access to an adaptive learning program so I plan to use it as part of the daily lesson so I can work with each grade.

For instance, I can have a whole group lesson for the 8th grade students while my 7th grade students are working on the computer and then I can have the grades switch so I can instruct the 7th grade students.  I also plan to have an opening and closing for the whole group together so they are not always separated.  Later in the week, I'll be sharing some websites that do some of the things that several apps did but are no longer available.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If it takes 48 apples to make one gallon of apple juice, how many gallons will 3900 apples make?

Warm-up

If there are 12 apple trees in your mini orchard and each tree produces 325 apples, how many apples will you get each year?

Sherlock Holme's Bike Story and Tangent Lines.

 

Did you know that there was a story starring Sherlock Holmes published back in 1905 in which he deduced the direction of travel from the tracks?  Did you know that some mathematician used this story to determine if he deduced the solution correctly.  

Sherlock was lying in bed, gazing at the ceiling when Dr Watson interrupted him to discuss a murder case. When Dr Watson described the tracks the bicycle left by the suspect, it captured Sherlock's attention and they were off the check the evidence. 

He had to determine which track belonged to the front wheel and which to the back so he could figure out the direction of travel. After reading a couple of new books and exploring differential curves in detail.  Due to the design of the bicycle, the two wheels are independent of each other but are at a fixed distance from each other and always move in conjunction with each other. 

At the end, Sherlock concluded the bicycle in The Adventure of the Priory School was going away from the school because the back wheel impression was deeper than the front wheel. He commented that the back wheel track wiped out parts of the front wheel track as it crossed over it back and forth.

The inaccuracy of his assumption is since the back wheel follows the front wheel, it always crosses the front wheel path.  Always. This is an important point.  The rear wheel always moves towards the front wheel because it cannot turn, it can only go in the direction of the front wheel. Now the important thing to remember is that at any instant, the direction of motion along the curve is tangent to the curve and this comes into play when you don't see the actual rear wheel line being deeper than the front or we can't see the real wheel path crossing over the front wheel one.  Furthermore, the length of the tangent line to the back wheel is always the length between the two wheels. 

If you were to create a graphic representation of the front and back wheels, you have the front wheel which looks like a standard sine wave and the back wheel is also a sine wave with a smaller amplitude but it cuts to the inside of the line representing the front wheel. Now if you were to place a tangent line on the curve of the front wheel, it does not intersect the other line within a bicycle width of the line but if the tangent line is on the back wheel line, it does intersect.  That is how you know which sine wave represents which wheel. If the bicycle is traveling to the left, the front wheel is to the left of the back wheel and if it is going right, the front wheel is to the right of the back wheel. 

So this whole mathematical premise is based on the tractrix formulas formulated by Newton back in 1676.  If a tangent line is drawn from the tractrix line a random point, towards the asymptote, it is the same length as the distance between wheels.  Now in real life, the paths of the back wheel doesn't always follow the tractrix formula because bicycles do not have the same wheelbase.  

This story is wonderful as a way to introduce this mathematical concept since everyone has heard of Sherlock even if they haven't read the book.  It is a great integration of Language Arts and Math. Let me know what you think, I'd love to hear. Have a great day.

Using Math Songs To Help Students Learn

Over the weekend, I ran into an ex-teacher who has started a foundation offering free music for certain concepts in math, science, history, and engineering/technology.  He maintains that if students are exposed to music containing concepts, they have a better rate of retaining the information. There is research that supports this thought and I know from personal experience, that if students can remember the song, they'll use it to help on tests. 

Before I discuss his stuff, I'll go into the idea of incorporating music into math to improve understanding and retention. There is research to indicate that students who are exposed to songs with mathematical subject matter tend boost test scores, enhance understanding of larger concepts, improves classroom climate, and makes it fun. This is because when information is set to music, students find it easier to remember. 

In addition, it has been found that the part of the brain that stores memories is also the same region that process both music and emotion so we can say that memories, music, and emotion are very closely linked in the brain. Furthermore, when mathematical formulas and concepts are set to music, our brains find it easier to remember the information.  If you listen to students, they can belt out the lyrics for their favorite songs and I know from personal experience they will belt out the lyrics to math videos I've played.

Research also shows that if you add movement in the form of dances to the math songs, many students will remember even more because it makes it even more fun.  So both the dance and music makes math so much more fun and on tests, students remember the songs so it makes them more relaxed and they are more likely to remember the concepts.

Although music itself doesn't teach students mathematics, it does help them remember and if it helps students gain a better foundation in elementary and middle school, they are going to do better in high school. 

So if you want to integrate more music into your math lessons, how do you go about it?  One of the first things you should do is to discuss the lyrics with your students so they understand the content.  You don't have to go over every single word, just talk about it to help familiarize them with it. Once you've played the song, ask students some questions about the song to help them understand the content.

The gentleman I spoke with is Tim Griffin who has a website containing songs for math, science, history, and engineering/technology.  He includes all the lyrics, the music, and notes including which standards are being met.  Although many of these songs are designed for younger grades, they can easily be used with older students.  There is a filter about halfway down the front page.  Check it out.

Another site to check is Songs for Teaching which has links to so many songs covering topics like multiplication, geometry and advanced topics. Unfortunately, not all the links lead to free access.  Some appear on CD's that have to be purchased or you can purchase the original song but many have samples and the lyrics so you can decide if it is for you.

The site I use most often is Youtube.com because it has so many math based songs. Number Rock is one channel with lots of different videos covering a variety of topics for a nice range of grades from elementary to high school. You'll find raps done by teachers, students, and math classes, along with some of the songs from people featured in Songs for Teaching.  Just put in the topic + song and you'll usually get a nice list.  

If you've never incorporated music into your classroom, check these sites out for inspiration.  Let me know what you think, I'd love to hear.  Have a great day.