Warm-up

If the tallest tank has a radius of 10.5 feet and a height of 32 feet, what is its volume?

Warm-up

The radius is .75 inches, what is its volume?

The Math Associated With Oceans.

The other day, I stared at the frozen river, wondering when it would thaw when I realized there is math involved in the oceans.  You might think that its quite a jump from river to ocean but it's really not because the river flows into the ocean a bit further down.  Once the river is thawed, traffic will begin again as locals head off to other villages, go fish, or wait for the barge to bring gas after a long winter.

1.  Although surfers do not use math, people who design artificial waves do.  They have to look at the math of waves such as the formula using wind speed, duration, and distance, applying the same mathematical period and amplitude found in trig, and other formulas that can be used to determine how the ocean might effect marinas, light houses, oil rigs, water parks, tourist resorts, and such.

2.  Another application of wave mathematics is when people decide to produce power using the energy of the waves.  It looks at harmonic motion, parabolic shapes to capture and focus water, the geometry of turbine engines, etc.

3.  Then there is the math associated with Tsunamis.  First there is the speed of a tsunami so people can predict the approximate time of its arrival, the predicted height of the wave, and the power of the tsunami wave.

4.  We can also apply mathematics to the hunting patterns of sharks, modeling of the ancient shark that has disappeared or  the force of a sharks jaw as it bites someone.

5.  Math can also be applied to ships at sea.  There is the math involved in how they stay afloat at sea,  the mathematics of the height of a ship above the water, ship stabilizers, sea sickness, stabilizing pool tables and swimming pools, and the geometry of ship hulls.

6. Finally the mathematics of surfboards.  I remember growing up in Hawaii, we knew when spring arrived because quite a few people disappeared to surf during the afternoons.  We see math in the different types of boards, the board to weight ratio, the geometry applied to the different parts of the surf board, the measurements of the various fins, the tools used to make surfboards, and so much more.


This site is great because it has detailed lessons on each of these topics. There are explanations, videos, illustrations, everything but the actual lesson with the objective etc but any teacher can put that together.  I know I will be using the wave material from this site for when I teach sin and cos waves in trig in a couple of weeks.

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

Battleship Trigonometry.

Apologies for not publishing anything yesterday but the internet has been extremely slow for the past few days.  When I tried to write this column, the internet basically stalled so bad, I couldn't even download my mail.  It looks like I'm back again.

I just got the latest issue of The Mathematics Teacher from the National Council of Teachers of Mathematics and on the front was an article on this topic.

I'd never thought of using trigonometry in reference to battleships.  The article starts with an explanation of armors and ships.  Since I know nothing about battleships, the explanation of "belt armor"(vertical plane or side) and "deck armor" (horizontal plane or top of the ship).  This is needed as the author goes on to show how an artillery shell might hit the belt armor or deck armor.

For purposes of the exercise, most factors affecting the shells ability to penetrate the armor is ignored and the penetration is set at 400 mm or the thickness of armor that a shell can pass through and cause damage.  In other words, if the armor is thick enough, the shell will not cause any damage, if not, the armor is damaged.

The idea behind the exercise is to use the path of the shell passing through the armor as the hypotenuse of a right triangle while the inner wall of the armor is the side and if a line is drawn between the shell's path and the inner wall, you get the other side.  Then just after the drawings are shared and before the math is done, students are encouraged to make predictions of will the shell penetrate either armor.

The second part is taking various values for range or distance from target, angle of depression, and artillery penetration (values taken from real life facts from the Iowa class battleships for the United States Navy 16 inch Mark 7 guns.  They use this information plus the definition of the sin ratio to determine the thickness of the armor and if the shell will penetrate the ship.  This math is similar to the math done by those who design battleships.

At the end, students should discover that the belt armor is more vulnerable than the deck armor even though it is thicker.  I didn't know this application of trig and since I've just started teaching trig ratios in my Algebra II class, I plan to use this activity early next class with those students.  I think its important to show students real life applications that do not seem contrived.

Please check it out for more details but I'm thrilled with this application and I'll report back with the results later next week, hoping the internet does not go down again.  Let me know what you think, I'd love to hear.  Have a great day.

Integers and Temperature

I love using real life examples of temperature up here in Alaska because we go down below zero.  Instant negative numbers with drops and rises so you are at -30 and drop ten degrees to a final temperature of -40.  Or it might be -30 and the temperature rises ten degrees to -20.

Unfortunately, this is not an easy concept to teach in many parts of the country since low temperatures may get down to 52 degrees.  I was in Hawaii, one winter, and the front desk worker was shivering because it was a cold 52 degrees F.

I laughed because that is warm to someone who'd just come from -40 or so.  In Alaska, we understand temperatures below zero but not everyone does.  We also understand that temperatures can effect the way things are built.

In many places, there are ice rinks which must be set up so the water spread over the area is frozen properly and the temperature maintained for weeks.  The temperature of the hockey rink has to be maintained between 17 and 23  degrees F (considered hared ice) which lowers the temperature of the surrounding air to between 50 and 60 degrees.  This temperature is sufficient so anytime water is spread over the existing ice, it will freeze.  On the other hand, the ice in ice skating rinks are generally kept between 24 and 29 degrees F for a softer ice that works best for ice skaters.

Many building materials react to changes in temperatures such as concrete. If the temperature drops too fast and freshly poured concrete is not properly protected, the water in the concrete freezes, creating ice crystals and the concrete looses strength.  In addition, steel can expand if the temperature goes up and it contracts if the temperature drops.

I see the results of soil being frozen during the winter at my apartment.  Someone in the past demanded they put solid skirting around the base of the building.  This meant the soil under the place didn't get as cold as that surrounding the building and its buckled causing the walls to separate from the ceiling.  I have gaps up to an inch which weren't there 13 years ago when I moved in.  Its only gotten worse over the years.  yes, occasionally they've tried leveling the building but it only lasts till everything freezes and the whole place starts pulling apart again.

What this all means is that if you are in construction, you have to plan ahead so you work at optimum temperatures for the material.  For instance, if you are pouring a concrete road in winter, you pour the concrete when temperatures begin to rise before covering it so it has a chance to set properly, otherwise it doesn't set properly and you've wasted money. On the other hand, if its the middle of summer, you might pour concrete at night when temperatures are much lower and have less effect on the setting of concrete.

There is another issue with extreme cold.  If you have a material that contracts while you are painting it or gluing it, either the paint or the glue could crack or wrinkle when the temperatures rise and the material contracts again.  This means anytime a contractor creates a quote for a project, they have to keep in mind the time of year, the temperatures, and how these temperatures could effect the timing of the project.

Just a bit on how temperature can effect buildings, construction, and materials while looking at it in terms of positive and negative numbers.  Let me know what you think, I'd love to hear.  Have a great day.

A Funny Thing Happened On The Way To Quitting.

This is the year I decided to quit teaching.  I am tired of an administration that does not communicate well.  I am tired of all the rumors that run rampant because our superintended is unwilling to talk to us, and I am tired of so many other things.

One of the first things that made it hard to teach was when the new iPads did not get released to teachers until mid-December.  When I finally got them, I couldn't use them all the time due to a limited band width.  Anytime they test online, which is quite often, we cannot use the internet at all.

We were supposed to get Apple TV's but only two classrooms have had them installed as of today.  There are supposed to be new computers available but no one has seen them so who knows what is going on with that.  I'm frustrated because I have to rely on worksheets more than anything else.

In addition, the state government is threatening to cut 25% of the education budget which will have a significant impact on things but no one is talking about it. What we've heard via the grapevine is they will not be replacing any of the high school teachers.  Instead they will just put the kids online rather than hiring new teachers.  The classes in the elementary will be bigger because they are going to make due with fewer elementary teachers.

As I said, rumors are flying and no one is giving out information so I turned in my notice back at the end of January.  Low and behold about two weeks ago, one of the other teachers told me I needed to apply to the village his wife is from.  He said it would be fantastic and awesome.  So I went ahead and applied.  Then I saw another ad for a position in a district I had worked for previously that had a great calendar for me and would allow me to do what I want in the summer.  I haven't really heard from either one.

Then another coworker encouraged me to apply for a math job in the district she just got hired in.  Ok, it means I won't be able to participate in an international conference in Ireland this year but it would be in a place where they are very much about using the internet and technology for differentiation and for support.

I sent off a cover letter and resume this past Friday and I have an interview at the end of the school day, today. It would be nice because I'd know at least one person there. She'll be in another location but its close enough to pop in once a month and I'd be in the hub with the district office.  It would be cheaper to fly in and out of there, then it is to get here because they are serviced by larger airlines.

I finally sat down and realized its not that I'm tired of teaching per say, its more like I'm burned out here working with students who think a "D" is good enough or they don't really care about school because their parents are alcoholic or strung out on drugs.  I need a new place where I can try with different students. Students who want something more than a "D".  Who are proud of themselves and their work.

So I'm looking elsewhere instead of quitting the field.  This will tell me if I'm just burned out here or if I'm ready to leave the field.  Let me know what you think, I'd love to hear.

Warm-up

The base of this pyramid hotel in Las Vegas has a base of 600 feet by 600 feet by 350 feet tall.  What is its volume?

Warm-up

The base is 116 by 116 by 70 feet high.  What is its volume?

These Are Not The Best Study Strategies.

 Yesterday, I looked at the best study strategies but there are two that should be thrown out completely and they are ones that I grew up with and I'm sure many of you did too.

First is rereading the material.  This is a very popular strategy with 84% of surveyed students using it but it does not enhance student learning because this method does not transfer the information into long term memory. 

This is one of those techniques, I learned when I was in high school but a lot of research has happened since I graduated and its one I tend to use when I'm making my own notes so I restudy the information.

Next is the use of highlighters to illuminate important passages.  This does absolutely nothing and according to one study, students who did this, performed worse on tests that required people to make inferences.  It is thought that when they highlight something they are not making connections with other parts of the text. 

I admit, this was a technique I never did because I hated messing up the perfectly white page with yellow.  The other thing is that if you ever bought used textbooks in college, they were often highlighted and the material highlighted was never what I thought important.  I'm sure many of you felt the same.

Summarizing the main ideas of the text appears to work if students are taught how to effectively paraphrase the ideas.  The key here is that students need to be coached in this strategy.  The training begins with students learning to summarize short paragraphs, before moving on to longer sections, eventually learning to do complete chapters.  The amount of training needed may make this strategy one that is not taught immediately.

This was never considered a study strategy when I was in school because its something you did in your English class, never when preparing for a test.  So its not one I ever learned.

Another technique requires students to visualize the content of the material they are studying.  Linking a visual with the content can help retain the material a bit longer but it only works with text that is too complex.  Unfortunately, it only appears to work for a short time after reading the material and the benefits do not last.  It is also harder for younger children to do because they often have trouble coming up with images for certain things.

The final technique is called keyword mnemonic where people link images together for words or idea. This often works well when taking a foreign language or working on vocabulary.  Otherwise, it doesn't help students for more than a short time.

I hope you learned something new in this today.  Let me know what you think, I'd love to hear. Have a great day.


 

The Two Most Effective Study Strategies.

We all know a way to study, especially when tests are upon us.  When I was in college, we highlighted, we read and reread the books, we deciphered our notes, all the night before in the hopes we'd remember enough to pass the test.  Since then research has been done to determine which strategies are the best.

One of the best ways to improve student performance is to provide frequent tests/quizzes rather than one for the end of the unit.  Students who used this method to study for their major tests, discovered they'd perform better than their classmates who used more traditional ways of study.  With the internet, it is easy to find all sorts of practice tests, either paper or online, to try.  As teachers we shouldn't be afraid of interweaving short ungraded quizzes throughout the unit.

In addition, a quizzing oneself does not just use traditional quizzes or tests but can include where the student hides information and tries to recall it or creates 3 x 5 cards or digital cards with questions based on the material they can use to test themselves. They should continue using these flash cards until they can easily answer the questions. As a teacher we can encourage students to leave space in their notes either to the side or on the backside where they can try a practice test based on the notes.  The same can be said of notes.  If a student keeps testing themselves on their notes until they can recall the material, they stand a much better chance of recalling the information during a test.

Next is distributed practice which in math means instead of doing all the same type of problem at once before moving on, they work one problem, then the next type of problem, then another type of problem until they've worked their way through all the types of the problem before starting again.  This has a much better chance of having students really learn the material.

This can be applied to preparing for a test by reading and rereading notes over a period of several nights rather than cramming the material in over a span of several hours the night before the test.  Research has found that if students take the time they'd spend cramming and spreading it out over several nights, they will learn the material better.

Although students might disagree, feeling they know the material well using traditional methods, research indicates actual learning takes much longer and if a child struggles, they are learning the material better.  The cramming method just has them recognizing the material for the moment but they have not had the time to properly transfer the material into long term memory

If you take time to teach study skills in your math class, these are the two that should be taught first because research shows they are the most effective ones.  Tomorrow, I'll look at those strategies that are totally ineffective.  Let me know what you think, I'd love to hear.

The Math Design Collaborative

 The other day, I was searching for something quite specific and ended up at the Math Design Collaborative from the Bill and Melinda Gates Foundation. 

The first thing I found were class outlines for grades 6 to 8, Algebra I and II, and Geometry.

These outlines begin with outlines comparing the material in grades 6, 7, and 8 or Algebra I, II, and Geometry and a list of the eight mathematical practices.  Then you get the actual list of content with formative assessments and how long each section should take.  In other words it has a nice pacing guide built in.

Once you get to the main part of the guide, it gives additional break downs for each lesson from a general introduction to the lessons, links to the appropriate material, tasks, books, and information on the formative assessments.  Everything you need for the lessons.  The best thing about the lessons, is they've listed the standards each lesson addresses so you don't have to sit there and figure that out.

Every lesson has tasks associated with them.  The tasks are all labeled so you know if they are designed for the novice, an apprentice or expert so you know who they are suitable for and if you need to add some scaffolding.

Under each grade level or course outline there is a list of each individual formative assessment. For instance in the Algebra I course, there is a section on linear equations with something on simple and compound interest rates where students compare a linear function with an exponential function.  The outline provides an overall view but when you click on the comparing interests link, it takes you to the actual lesson module.

Although there are links for every section of the lesson, not all of them work correctly. The one link I found that did work, was the one that sent me to The Mathematics Assessment Project lesson on representing Linear and Exponential Growth.  This comes with the necessary resources including the lesson and a power point. 

The other problem with the detailed lessons is they often refer to pages that are in the material from the Mathematics Assessment Project and if you don't know that, it can make you wonder where the material is.  Otherwise, the information is there to follow step by step from pre-assessment through to follow-ups. 

I like the whole concept because I don't always have the time to write a lesson from beginning to end.  I found a nice lesson on classifying systems of equations I can use next week in my Algebra I class. 

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

Times in Music

I was at dance practice last night but its wasn't the usual dance practice you might be picturing.  It was a Cup'ik (Chewpick) practice.  That means they used a standard 4/4 beat with a slight variation but it was usually 4/4.

In the middle of doing a song about two men who went whale hunting one day and ran aground on a sand bar, my thoughts flittered to this piece that was rather complex.

It was like 2/4, then 3/4, back to 2/4, then 9/8 thrown in and it was crazy.  It was extremely easy to get lost in if you were not paying any attention.  In the middle of all this, I realized it's all a form of math.

In a sense, time signatures are a fraction where the top number or numerator indicates the number of beats in one measure of music while the bottom number or denominator indicates the type of note.  So a 4/4 means there are four beats in a measure and each note is a quarter note.

Not quite the same as parts of a whole but not all fractions are parts of a whole.  Look at ratios which are comparisons and time signatures which provide a different type of information. Neither one is really considered Part of a Whole.

Furthermore, musical notes and rests are the visual representation of the time signature.  Any musician can look at the music itself, count the number of notes, look at the value of each note to determine the time signature without looking at it.  This is much like mathematics where its numbers are like the time signature and the representations of those numbers are the notes.

There is a mathematical equation for determining the note value based on the number of flags on the note.  The equation is 2^(-2-n )where n = the number of flags.  If its a whole note, it is 2^0 power or 1 beat where as if it has 4 flags, it would be 2^(-2-4) = 2^-6 or 1/64.

In addition, most pieces of music have repeated sections in them denoted by a repeat sign.  Most pieces of music have some sort of repeating pattern just like math has.  I know I always take a few minutes to look for the repeating pattern in the music before playing it so I don't miss any.

Then you have the vibrating sound waves that are tuned to a certain standard of Hertz.  Middle C is found at about 262 Hertz while other notes are at a different frequency.  If the instruments are not tuned properly, they some times sound off.

Even Fibinocci contributes to music. If you look at a piano, you can see it with the 13 keys that make up an octave.  Five of the keys are black and eight are white and the black keys are grouped in a group of two followed by a group of three.  2 +3 = 5, 5+8 = 13.  Usually a scale is made up of 8 notes where the foundation of a chord is based on the 3rd and 5th notes.

It is said that Mozart based many of his piano sonatas on the golden ratio.  In addition, the violin's parts are also based on the golden ratio and provides the basis for saxophone mouth pieces, speaker wires, and even used when designing certain cathedrals.

So math is found in so many places in music.  Let me know what you think, I'd love to hear.  Have a great day.

Angles, Skiing, and Snow

 Skiing is one of those wonderful winter sports that people with drive a couple of hours to do.  I tried it once but I proved to be quite unsuccessful in that I manage to go down a hill backwards.  The only reason I didn't go very far was this beautiful tree I ran into butt first.  I've never been again since that day.  Besides its intimidating to watch a 6 year old whizzing down the advanced slope.

The cool thing about skiing is it involves math, grades or slope, and so much more math.    In skiing, they refer to slope as Piste which is French for the marked run down the mountain.

First off, slope in downhill skiing is spoke of as percent slope also known as grade.  A 45 degree angle is actually a 100 percent grade.  If you can visualize it in your head or draw it on graph paper, it shows up as a 1/1 slope or a rise of one foot over a run of one foot.  In addition to slope, skiers must keep in mind that snow no longer sticks when the slope is over 75 degrees.

Furthermore, they must keep tract of the type of snow because the type of snow determines how fast a person can go down the hill. If the run is made of up ice, the person will ski harder than on powder snow.

In general the easiest trails have a slope between 6 and 25 percent while the next ones up are between 25 and 40 percent.  Most ski resorts have the majority of the trails in this range.  The more expert skiers prefer trails over 40 percent because of the challenge.  The hardest trails have slopes well above 40 percent and lots of obstacles.

Now for the question of how the type of snow effects skiing.  If it has not snowed in a while and people have compacted the snow by skiing repeatedly,  it becomes ice.  Ice is not well liked by many skiers because they cannot dig their edges into the ice and they don't have much control.

If the snow is soft, it is easily moved around by skiers and it allows them to dig their edges in so they have the most control.  In addition, if they fall, they won't be hurt because the snow pads them.  If the temperature gets too warm, the snow will melt during the day and refreeze during the night when temperatures drop.  This means the run can be quite icy in the morning but gets slushy later on as the temperature goes up.  Slushy snow lacks structure, has lots of ice, making it harder for the skier to control their run and they can get ice burns if they fall.  When snow is wet, the water can create a vacuum under the skies, they don't slide easily.  Its like snowing with breaks on so the skier finds it quite hard to go downhill.

So the type of skiing one experiences when going downhill depends on both the grade or slope, the type of snow, and the temperature.  Imagine the types of graphs students can create from this information.  They can graph the degrees of each type of slope to see how steep it is.  Then they can create the grade such as 1/1 for the 45% angle to meaning to the percent. They can also create a graph based on the temperatures for the different types of snow for skiers.  Finally, they can compare and contrast the classification of green, blue, and black slopes based in the United States or in Europe.

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

Happy St. Patricks Day

Warm-up

It took the skier 352 seconds to go down the hill.  If the skier traveled at 52mph, how long was the run?  

Mathematical Communication, What?, How? Why?

As teachers, we tend to spend more time teaching students content and  process than we do to communicate so when we ask them to explain their thinking, we get those blank looks, or I don't know, or shrugs.  We need to spend more time on this but if you are like me, teaching students to communicate their mathematical ideas was not part of my teacher training.  It has been shown that effective communication is needed for rigor and deeper understanding but most teachers have never taken time to instruct students in how to do it.

In addition, mathematical communication helps improve learning but gives students some real life experience similar to what they might face once they are out of school.  When asked to explain, it helps if they can formulate their thoughts into words or representation necessary to express them to others in a coherent, understandable manner.

Furthermore, mathematical communication helps deepen their conceptual understanding while expanding and refining it and it allows them to meld their ideas with others and it requires careful thought.

 

The process requires that students need to express themselves orally, using representations, and in written form.  One way to do this is through the use of literacy strategies usually used in English such as word walls, modeling, revision, shared writing, and examples.  All ways to show students how they need to talk and write math.  

As they prepare to share their thoughts, they end up reviewing what they know about the topic, make new mathematical connections, are able to organize their ideas, become able to decide how important the idea is to the topic, are able to share their ideas with others, learn and use the appropriate vocabulary, create a cohesive idea to share their ideas, and express the ideas verbally, written, or through some form of representation.

At the same time, they have to listen to others ideas, compare those ideas with what they know and believe, create new knowledge and add it to their own, decide how to respond and with what information, then deliver the response.

This is an ongoing process that repeats to refine their understanding continually.  During the process, students often discover their own weaknesses and misunderstandings.  Furthermore, teachers can use these communications as a form of assessment to help pinpoint student deficit. 

One important step is to create a classroom that invites communication.  Arrange the classroom so students can face each other because it encourages communications and have a carpeted area where students can gather in groups.  Place whiteboards so students can use them to illustrated their ideas.  Make sure students understand that errors are a normal part of the learning process.  Let them know your expectations for verbal, written, and representative forms of communications so they know what to do.  Finally provide lots of opportunities for mathematical communications using authentic contexts.

If you are not integrating communications into your daily class, you should start.  le me know what you think, I'd love to hear.  Have a great day.

Happy Pi Day!

It is that time of year again when we celebrate the beauty of a ratio without whom we could not find the area of a circle or the volume of a cylinder or cone.  It is found in nature but its also a number that government has tried to regulate via law or has been adopted by a military unit.

I am proud to announce that I have several T-shirts that deal in some for with that wonderful create we call Pi.  My students think I'm crazy but I'm just into math.

Its also a number that has been around for a very long time.  The ancient Babylonians took three times the square of the radius for the value of Pi but documents have been discovered showing they assigned a value of 3.125 for it around 1900 B.C.

Another document, the Rhind Papyrus from Ancient Egypt shows the Egyptians calculated Pi to be around 3. 1605 and this is still a fifteen hundred years BC.  Then around 200 BC, Archimedes, the famous mathematician, calculated the value of pi by using the Pythagorean theorem to find the area of a polygon inscribed inside a circle and a polygon with a circle inscribed within it.  He was able to show the value of Pi was between 3 1/7 and 3 10/71.

Even the Chinese had a mathematician who worked on calculating the value of Pi without being aware of the work done by Archimedes.  Zu Chongzhi came up with a value of 355/113 for Pi but we are not sure how he did it because much of his work has disappeared over time.

It took till 1706 for someone, namely William Jones, to assign the Greek letter Pi, the same one we use today to represent Pi but it did not become popular until Euler began using it in 1737.  Apparently, the letter was taken from the Greek word for perimeter. During the same century, a French Mathematician used probability to calculate the value of Pi.

Prior to computers, William Shanks, in 1874, calculated Pi to the 707th digit but only the first 527 were correct.  It wasn't until 1945 that D.F. Ferguson calculated Pi to 606 digits correctly.  In 1947, D.F. Ferguson used a table calculator to find the digits to 710 places.  With the computer, it became easier to calculate pi and has been calculated to over the 10 trillion places.

In addition, you'll find pi popping up in books, television, movies such as the Twilight series, in the Fox Trot series with the two young men, the Simpsons, cheers at MIT and Georgia Tech, colognes, and there is even music based on the digits of pi.  Its found in nature, in architecture such as the Pyramids, and its all around us.

I attend conferences where people wear t-shirts with Pi on it.  I have several and am proud to wear them.  Let me know what you think, I'd love to hear.  Have a great day.

Coding + Math = Improved Math Scores

 I so want to integrate programming into my math class but right now the total bandwidth for the school is so small that it doesn't take much for the whole thing to go down.

Teaching coding in math can have some wonderful benefits for students.  Many people believe students who code should already have good math skills but research indicates that coding can help students develop strong math skills.  In addition, it is fun.

The cool thing about coding is that as they learn to code, they are simultaneously developing problem solving skills,  practice computational skills, and learn algorithmic thinking.  Often times, coding can help students visualize the more abstract concepts.

As a student creates a program, they have to use problem solving skills to write the code.  If they move their avatar forward, they determine the number of steps, if it has to jump over anything, or if it needs to turn around.  All of this requires thought.  If the program does not work properly when run, the student needs to analyze where the error occurs in order to correct it.  All, real life problem solving.

Imagine, this fun activity develops critical thinking and problem solving both skills are necessary to mathematics.  In addition, its hard to get students to "make sense" of problems while "persevering" on tasks that do not grab their interest but coding easily does this.  It teaches them to look at a variety of ways to solve a problem.  It may sound simple to have a character walk across the screen without tripping on running animals but it all takes a lot of thought to do.

Furthermore, coding requires students to understand counters via loops, variables which can represent different values depending upon what is needed, integers are used in so many things from commanding your avatar to walk so many steps to causing something to rise or fall.  Many times, the coding commands require the programer to use x and y values to locate the avatar on the background.  The x and y values can be positive or negative depending on where the avatar needs to be.

Coding also opens us discussions on mathematical concepts such as having students write a program to determine if a number is even or odd.  We know that even means it is divisible by two but what does that really mean?  If you stop and look at it visually, divisible by two means there are no remainders which means zero left overs.

In addition, it makes rounding more realistic.  The standard way that rounding is taught is the old if its four or under, you round to 0 but if its five and above you take the number up but think about reality.  If you are trying to pack computers in boxes for shipping and each box takes 15 computers, how many boxes will you need if you have 61computers.  you have one left over, so you would only need four boxes to ship the computers but then you have not sent them all.  This illustrates the need to round up.

If you want to integrate coding into your class but you are not sure how to start.  Scratch is a language that can be downloaded onto school computers for free.  This site from Harvard has lots of good information on using Scratch across the curriculum.  The pdf begins with links to general articles on integrating Scratch across the curriculum before looking at using it in specific subjects.

The math part has a section on resources, actual math activities, followed by suggested projects for grades 3 to 8.  I looked at some of the suggestions for the younger grades such as making a simple calculator and that looked like fun.  I think I'm going to download Scratch, the information on projects, and play with them myself.

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

The Cost of Daylight Saving Time

It's the time of year when people arrive at work, looking quite groggy and they head straight for the coffee pot because the time change threw their internal clocks out of wack.  Most of us can't figure out why we do this other than the government wants it done.

Unfortunately, there is a cost associated with this time of year when we go onto Daylight Saving Time.

It is well known that when clocks are turned forward, there is an increase in heart attacks, workplace injuries in certain sectors, and increased car accidents.

Furthermore, productivity decreases in the spring when clocks move forward an hour.  One study indicates the amount of energy used also increases during this same period.

A common thought is that with the extra hour of daylight, people are likely to stay out and spend more money, helping businesses  but according to studies this is not true.  The increase for small businesses appears to be marginal at around a 0.9% increase in spending via credit cards but when the clocks turn back, the amount drops by over 3%. It has been estimated that this can cost the economy between $430 million and $1.7 billion in when clocks are changed twice a year.  The second figure is based on the idea that time is money.

Now for some very specific stats in regard to Daylight Saving Time.  In the mining industry, the shift results in an 6% increase in injuries and the numbers of workdays lost increases by 67%.  In addition, the amount of surfing for entertainment and related categories increases between 3.1 and 6.4% on the Monday immediately following the change.  This means people are working less by that amount.

According to mathematical predictions, heart attacks on the Monday immediately following the time shift increases by 5%.  Furthermore, we loose about 40 minutes of sleep in the first day on Daylight Saving Time which can hurt our judgement due to sleep deprivation because the change throws off our internal clocks and makes us restless.  It has also be noted that judges hand out sentences that are 5% harsher than usual on this same Monday.

Furthermore there is some evidence that the change also effects SAT and other test scores and the stock market.  So as you can see, there is an economic (mathematical) cost involved in moving the clocks one hour ahead.

Its easy to go to the internet to find averages for heart attacks on any one day, or mining injuries, or sales and apply these figures to them to see how the numbers change.  Students could also look at these same figures for the month before and month after, or even week before and week after to see how the numbers change before preparing the information in graphical form.

This would be a good topic to have students create a news report, a presentation against going on Daylight Saving Time, a informational pamphlet, or something similar.  I hear we shouldn't have students make posters, etc anymore but I believe these types of things are worth doing if they are put in context and the suggested activities fall within that realm.

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

50% Really?

 This past Friday, we received an e-mail from the superintendent informing us about a change he made to the grading policy back in November.  We must now give students a 50% if they are enrolled in the class.

This change to board policy was made without asking any teacher what they thought.  It was not communicated to the parents.  It was not even read out loud at any board meeting per board policy.  It was unilaterally changed.

I had heard it was happening but I didn't worry because he'd never informed us.  The principal occasionally mentioned it but more in the context of "The super is planning on doing this."

The idea is that a student is given 50% automatically for the quarter so that all he needs to do is manage a 70% for the next quarter, so he finishes the semester with a 60% and passes.  Unfortunately, it might change the student mindset into the "I'll do well the first quarter, then do nothing the second and I'll still pass."

Unfortunately, the only "research" supporting this stance as far as I can tell is published by the group who believes in this stance.  I cannot find any real research that delves into this topic.  I've found antidotal information that shows this does not work.

I can tell the super got this from somewhere on the web because it mentions AP classes.  The majority of our students in high school are struggling in most classes because they are well below grade level when they arrive to 9th grade.  Our better students tend to apply to Mount Edgecomb boarding school which offers a much stronger and on level program.

The super did provide a form for teachers to file to get a student exempted from the 50% rule but you have to show the student did not attend, did not try, you spoke to the parents and even then, its up to the administration to agree to the student receiving less that 50%.

I do not think this new policy is going to change students who choose not to work into students who will work all the time.  I also know that several high school teachers already give students so many chances to redo their work and improve their grades.  Basically, as long as they are willing to try, they will pass.

Furthermore, the students who try, even if they do not pass the quarter, have at least a 50% so this new policy does not effect them.  It only effects the students who do little to nothing in class or who are gone most of the time.  These are the students its meant to help but until they are ready to work, it really won't help.

Let me know what you think, I'd love to hear from people who have experienced this type of grading policy.  I'm curious as to whether it really works and what type of students does this policy benefit.  If you have any information, please let me know.  Thank you.  Have a great day.

Warm-up

Name 5 things that have a unit rate.  Explain the units involved and the context of the unit rate.

Warm-up

Name 5 situations with both positive and negative numbers.  Explain the meaning of the positive and the negative within that situation.

Google Celebrates Olga Ladyzhenskaya

 Yesterday, Google celebrated Olga Ladyzhenskaya, a Russian Mathematician.  She was born in the 20th century and died in the 21st  but she had an impact on the subject.

Olga was born on March 7, 1922, in a small village in western Russia.  She developed a love of math due to her father who shared his love with her.  He taught mathematics.

Unfortunately he passed in 1937 when the Soviet government declared him an enemy of the state, arrested, and killed him.  In 1939, when she graduated from our equivalent of high school with honors, she applied to Leningrad University.


Due to her father, they denied her so she applied to Pokrovski Teacher's Training College before taking over her father's job teaching math.   Supposedly, she talked her way into the teachers college before her application papers could be transferred from Leningrad University.  From the information, I've found, she appears to have been admitted into Moscow State University in 1943.  Although she married another mathematician in the 1940's the marriage did not last long because he wanted children and she preferred devoting herself to her work so they parted ways.


Once World War II ended, she was able to transfer to Leningrad University obtaining a Master's degree before earning her first PhD.  At this point, she transferred back to Moscow State University where she earned her second PhD in 1953. Upon earning her second PhD, she got a job at the Laboratory of Mathematical Physics at Steklov Mathematical Institute in Moscow in 1954.

Eventually, she lead the laboratory and while there, she wrote over 250 papers.  Her mathematical works influenced weather forecasting by refining equations used to describe cloud motion and weather patterns, aerodynamics, and equations used to describe the motion of blood in cardiovascular science.  She is best known for her work on the Navier - Stokes equations which mathematically describe the motion of viscous substances.

In 1956, the Soviet government officially exonerated her father due to a lack of concrete evidence of the crime.  Unfortunately, this made it so she could not easily travel outside of the Soviet Union.  She only every made two trips out.  The first in 1958 to attend the International Congress of Mathematics and again in 1988.  Once Communism fell, she began traveling more.

During her lifetime, she wrote multiple books, and was always a leader in partial differential equations and mathematical physics.  Its amazing that her work is still influencing areas today.  One really interesting thing is that she suffered from an eye problem that required her to use special pencils.

Due to her work on that and differential equations, she received the Lomonosov Gold Medal in 2002 after she'd been denied the Fields Medal in 1958. The first women to win the Fields Medal, did so in 2014.  Olga died two years later at the age of 81, on January 14, 2004 just before she was scheduled to depart for Florida where she planned to finish a paper.  At the time of her death, she had five years of research she wanted to work on.

Let me know what yo think, I'd love other.  Have a great day.

Trying Something New.

My 9th grade math class is extremely low and very unmotivated.  The only time they really get excited is when we play Kahoot, Jeopardy, or work on a games based website.  Unfortunately, I have to give tests to monitor their progress in addition to using the results of games based websites.

This group shuts down and gives up easily.  They lack a lot of motivation and do not do well with regular tests so I've written a partner test.

I divided the class into groups of two students.  One student gets version A while the second student gets version B.  Each version has different problems but both problems have the same answer.  One student might have 24 x 36 while the other 54 x 16 yet the answers should match.

If the answers do not match, they know instantly something is wrong.  I chose to do it this way so they get immediate feedback and they can check their work as they progress through the test.  Furthermore, it slows them down to really stop and check their work each step of the way.

In addition to giving immediate feedback, it means they have to communicate to explain what they did and why they chose to solve it a certain way. This method also requires them to look for mistakes and helps them build perseverance in a safer situation.

I've known teachers who think of this type of test as cheating because students are not doing this on their own.  I'm more concerned with students learning the material than them "proving" they know the material.  I'd rather give a test like this to students who have little motivation.  I'm hoping they get more confident and are willing to work more independently.

What's fascinating is the lack of research or information in general on this topic.  I look up partner tests and get all sorts of on-line places you and your "partner" can go to see if you are a good match.  Even when I added "math" to the mix, I still couldn't find anything dealing with partner tests.  I even went so far as to type in "Giving math tests to two people, each test has different problems but they have the same answer" and ended up with all sorts of references to taking the Praxis or how they were scored.

This is apparently an area that has not had much written on it.  I wonder if people do not think its worth it or if its been disproven.  A partner test does provide some wonderful information via both the finished product and by observing students working on the test together.

I'd love to hear from others. What do you think of this idea?  Let me know.  Have a great day.

7 Ways to Increase Mathematical Reasoning.

It's March, and we've started going over practice questions for the states required standardized test.  I give one problem a day, and ask students to explain their answer.  Unfortunately, they are still at the "I guessed." or "It popped into my head." or "I'm Shaman.".  I've tried to explain none of those really tell me anything because there was no real thought behind it.  No real reasoning.

This is the first way for students to improve their mathematical reasoning.  When they explain or justify their answer, they are able to examine the logic used during their thinking.  It is important to have students show their thinking process from start to finish be it verbally during exercises or in written form for daily work or even tests. Every time they show their work, they are communicating their thinking.

2. Use geometric proofs or some sort of two column proof.  In the first column, they write down what they are given, then what they suspect while in the second column, they explain why each statement is true.  Doing geometric proofs in this format force them to look at the small steps in solving it.  This is another way to help students see their reasoning.

3. Have students work together because it allows them to justify their thinking to each other.  In the process they can analyze and critic each other's thinking.   Set up "Brain Talk" where students use modeling, verbalization, or other ways to show their understanding and justify their position.  The teacher may need to ask questions such as "What is the same?"  "What is different?"  What do you know?"  to help them get their thinking going. The important thing with this discussion is students have to feel safe and they need a chance to come up with hypothesis and solutions.

4. It is also necessary to come up with agreed upon mathematical terms across the grades because it leads to less confusion.  This makes it easier to for students to continue developing their reasoning.  When I was in school, they used the term "Borrowing" when you needed to "Regroup" as they say in today's math language.  It is also important to encourage students to use mathematical language when they explain their thoughts because the more precise they can be, the better they understand the concept.

5.  Take time to have students look at problems done incorrectly and identify the mistake.  This process adds to developing student reasoning because it teaches them to really look at the process and numbers used to solve the problem.

6.  Encourage students to find two or more ways to represent any problem since its important to "see" things in a different aspect.  This helps students move to a different process if the first choice does not work.

7.  Encourage students who struggle.  Let them know that struggle is a normal part of learning math and developing their reasoning.  The struggle is when they develop their reasoning and as they work on solving problems, their reasoning improves and it becomes so much easier.

You don't have to implement all of these at once but use one or two to start with on a regular basis till students are more comfortable with showing their understanding.  Let me know what you think, I'd love to hear.  Have a great day.

What is Mathematical Literacy

Have you every stopped to look up mathematical literacy? Have you determined what elements you need to cover for your students to be mathematically literate?  Well, go no further because this answers those questions.

For a student to be mathematically literate they need:

1.  To be fluent in basic facts and computation.  It is important for students to be fluent in their basic facts because higher level math is built upon those basic facts.  In addition, fluency indicates that they've stored the information in their long term memories where it is easily accessed.  This means, they have freed up space in their working memory for higher level mathematics and to increase their problem solving abilities.

When a student is not fluent with the basic facts, they often get confused when working through more complex problems and are often lost.  This may be why my students who have to use their fingers to add or multiply often are not sure what their next step is.  These students struggle working their way through the process and often give up.

Furthermore, when they are not fluent in their basic facts, they focus on the calculations and often are unable to finish the longer assignments.  This can extend to other topics such as science or geography because they are focused on completing the math rather than seeing the whole topic.  One last thing, a students fluency in the basic facts can determine how well they do later in life.

2.  To learn math concepts beyond arithmetic.  It is this that helps students learn number sense so they know if an answer they found is reasonable or off base. When a student has number sense, they are able to think more flexibly and they become more confident.  When they lack number sense, they also lack the basic foundation needed for simple arithmetic.

3. To connect math to other subjects.  Its important for students to see that math is used in other subjects such as science, geography, architecture, business, and other things.  It becomes so much more relevant when they see math outside of class.

4. To reason mathematically. This means that students can follow arguments developed by others and create their own to prove or justify their answers.  A student who can reason mathematically can also look at questions and propositions in different ways, create and test hypothesis, figure out counter examples, draw conclusions, and figure out different ways to approach problems when stuck.  In other words, it helps us make sense of the world.

One way to help develop this is to begin with an open ended problem or an exploratory exercise so students have experienced the concept before presenting the theoretical segment.

5. To communicate using graphs, models, symbols, and language.  This is a way for students to exchange ideas and knowledge using both spoken and nonverbal methods.  It is important to help students develop this ability with modeling and practice.  Its a bit different communicating mathematical ideas than it is ideas from literature.  

In addition, when students are able to communicate using graphs, models, symbols, and language, it improves their understanding of mathematical concepts by melding their ideas with others.  It expands and refines their understanding.

6. To solve problems confidently.  When a student is able to solve problems confidently, they are more sure of themselves and open to learning more.  They are not restricted by their lack of ability but able to move forward to and be willing to try problems that may be a bit more difficult.

So know you know the six parts of mathematical literacy.  Let me know what you think, I'd love to hear.  Have a great day.

Cost Higher Than Normal

 I'm going to spend time discussion the cost involved for everyone of loosing cell phone, landlines, and internet signals to everyone involved because it is math.  Last year, we had issues with service for two months and they refunded my internet costs for two months.

I haven't spoken to them yet but things finally came up again Sunday Evening.  So here is some of the things that cost the village and the company.

1.  With no communications, the planes could not come out.  They lost money due to not being able to travel to the village.  They couldn't bring the cargo or mail and would be getting even more backed up.

2. No communications also meant that the phone company owes us for the time the landlines, cell phones, and internet service was down.  I'd say it was 4 days for cell phones and internet service and 2 days for landlines.

3.  There is also the cost for sending one employee by snow machine over to SB which had not lost service.  That includes gas, and his time.

4.  The company also had to pay to send a couple of technicians and a part out to SB to go fix the relay tower that had gone down.  Since it took a couple days, it meant they had to also pay for places for them to stay, etc.

5.  They had to arrange for the internet signal to come up at the school.  It wasn't as easy as turning a switch to shift the source directly to the satellite because last year, they used  a piece they didn't have this year.

6. The clinic had to use their satellite phone to contact Bethel Hospital in case of emergencies but no one could call the clinic for appointments since nothing worked.  If it was an emergency, they might drive to the clinic and hope they were open, otherwise, no one went.  This means some people might have gotten sicker and it would cost more to heal them.

7.  Also due to the break down of communications, no one knows if their paychecks actually got deposited into accounts.  It is possible for people with automatic payments, they might not have the money and could end up paying late or bounce fees.

This all adds up to be quite a bit of money.  I don't have the actual amounts but I thought I'd address it this way since it gives people something to think about.  Most people do not experience this much of a communication break down.

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

Off the Grid

Everything, the cell phones, land lines, and internet went out on Wednesday night.  The land lines and limited internet came on last night but they are shaky at best.  No cell phones for the immediate future.  I had to pop over to work to get this because its the only building in town with internet.  I hope to be back to normal soon.  I will update everyone Monday.