Real Life Inequalities.

  I decided it was time to have my students do  bit of the work so in Algebra I, I began one variable inequalities by having students find real life examples and write them on the board.

We run into them all the time but we don't connect the situations with being inequalities.  One such situation deals with posted speed limits.  You can go as fast as you want up to that speed but not over it or you might receive a ticket.

Another two situations involve credit cards.  The first is with the credit limit.  A person can charge up to that amount but no more so the inequality might read x < $10,000.  The other inequality is in regard to the minimum payment.  That is the least amount a person can pay but they are welcome to pay more.  The equation might look like x > $122.30. 

Twitter uses an inequality when they restrict messages to 140 characters while many contact forms might restrict the message to 500 characters.  Then there is the issue of fundraising.  Most groups set a minimum amount they hope to raise through selling candy, cookies, or a raffle.  Although they hope for minimum amount, they will not stop there.  Instead, they will continue accepting donations.

There are some jobs out there which require you to sell a minimum amount in order to receive additional funds.  It might be you must sell over $8000 worth of computers before you get a bonus.  This would be another real life example of an inequality. 

In addition, there are other examples such as elevators which have a maximum weight load usually couched as a maximum load of 9 people or 1000 pounds.  Check the bridge signs and they might say have a restriction on maximum truck weight or a height restriction so a vehicle must be under 7 feet off the ground.

There is also the minimum travel time listed when driving or flying.  Out here the minimum is 45 minutes from Bethel to the village if everything is great and you get the fast plane but it could take 2.5 hours if you end up going to the two other villages first.

Sometimes, when we find something we want to buy that special pair of shoes, or dress and we have to plan the amount of money we need to save each week.  Or when you buy a house or a car, you have a minimum payment to make each month.  My mother always said that if you pay $100 extra each month, you'll pay your mortgage off much sooner.  I've always kept that in mind.

So many different possibilities, all provided by my students the other day.  It allowed them to find the real life applications before we began studying the topic so they can build a solid foundation.  Later in the week, I'm going to give them an inequality and they will have to write the situation to go with it.  I'll let you know how that goes.

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

Real Life Parabolas.

  I can tell you the equation for a parabola since I teach it at least once a year but I have trouble applying it in real life because most of the textbooks I use only provide theoretical situations, not real situations.

We observe parabolas in real life when we watch a ball thrown because its natural trajectory is parabolic.  This doesn't matter if its in baseball or basketball.

You see the parabolic shape in reflectors, flashlights, and satellite dishes.  The reason the telescope was built in Arecibo, Puerto Rico is because there existed a hole which perfectly shaped for it. If you look at car headlights, you'll see they are backed by a wonderful parabolic shape which causes the lights to shine in its pattern. In addition, spot lights also use the parabolic shape to project light outwardly.

Even the curved parts of the suspension bridge have a parabolic shape as the cables are put under pressure.  In addition, if you look at water fountains, the water is thrust up and  when it reaches a certain height, it comes down in a parabolic shape. Of course, you use the parabolic shape when launching birds in any of the Angry Birds games, or when launching the pumpkins at the pumpkin chunking competition.

What about the McDonalds eating establishments with their double arch made out of two parabolas joined to create the M.  If you've ever looked at roller coasters, the whole ride is made up of multiple parabolic shapes hooked together to form a pathway for the cars.

Even skis have started having a parabolic shape carved into each side because these cuts make it easier to turn the skis due to a shortened turning area.  Then there is  the beautiful banana that is found in the shape of a parabola.   Just look at the world around you.  The slinky as it crawls down the steps forms a parabolic shape as it moves from step to step or the path of jumping fish or dophins as they rise out of the water before diving in.

I think its important to provide students with an opportunity to research these items and then figure out the actual equations to describe the specific shape. My college prep will be studying parabolas soon and I plan to have them research real life parabolas before beginning the mathematics behind them.

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

Avalanches and Math

  Avalanches occur anywhere there are slopes and snow.  Some avalanches occur in areas with little to no human interaction while others occur in populated areas.  Years ago, I took a class on the physics of avalanches and there was more to it than I'd ever imagined. 

Its interesting that there is not one type of avalanche.  There are many depending on the type of snow, the snow pack and other factors.  For instance there is a snow sluff or point release avalanche which involves new snow being released from a point and spreading outwardly as it goes down hill.  There is a slab avalanche which is when bonded blocks of snow move and are among the most dangerous type of avalanche.  Although about eighty percent of slab avalanches happen on a hill with a slope between 30 and 45 degrees, the most common slope is 38 degrees.

There is also math involved in predicting the chances of avalanches happening in certain areas due to populations, etc. One way of learning to predict possible avalanches was done by several scientists who discovered by dropping ping pong down an Olympic Ski jump,  they got a better idea of how powder snow reacts.  Apparently, they are light enough to be caught up by the air in the same way as powder snow avalanches.

As a result the scientists were able to develop simple models for momentum and for the volume of snow fall.  These models are being tested in real situations in Switzerland. Powder snow avalanches can reach speeds of one hundred miles per hour and can easily bury people.  If you'd like to present activities in your classroom that would give students a flavor of working with avalanches, there are these:

1. Discovery Education has a lovely activity which allows students to discover the angle needed for avalanches to start moving. It does not require a lot of materials but what it does require can be easily gotten, especially if planned for ahead of time as it uses pebbles, sand, talcum powder, and marbles.

2. PBS has this activity to help students understand more about how weak and strong layers of snow lead to avalanches. It uses flour, sugar, and mashed potato flakes to illustrate how it happens. 

3. This activity has students use different substances to see how they form heaps. The heaps are then measured for area, height, and angle before being graphed to visually compare results.  Students are expected to reach conclusions on the type of event after researching different kinds of avalanches.

I love it when I can teach a topic that extends beyond just the math classroom.  There actually is a lot of math involved but some of it is a bit advanced for my students especially when  looking at the turbulent flow of two component mixture used to describe the density and volume distribution.  Or the avalanche release and flow parameters.

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

Off the rest of the week

i received a call Saturday from my family telling me my sister was in deathly condition in the hospital.  I grabbed a flight and made it so I could spend time before she passed on. I will be back Monday after the service and other things.  Thank you for Understanding.

The Math Of Skiing

It is winter time and many people head off to the local ski resort to spend time out on the hills.  They love flying down the hill as fast as they can and once they reach the bottom, they head up again.

I tried it once but between being intimidated by 6 year old's whizzing down expert slopes and skiing backwards down a hill, I've never been back.  If you are wondering, the only reason I stopped was that a lovely tree stopped me when I ran into it butt first.

The only skiing we have around here is cross country skiing. There is quite a bit of math associated with down hill skiing in terms of the hotel, cost of skiing, and the course itself.  People have to calculate the cost of driving to the resort, the cost of the rooms, the cost of renting equipment, lift tickets, etc so they can determine the amount of money needed to finance the trip.  The cost may not be as much if a person skies down a local hill.

In addition, the ski resorts use mathematics to determine the current amount of snow and can predict possible snow fall based on mathematical modeling.  When predicting the amount of snow, they have to know wind direction and speed because that gives a good idea of where the snow will fall. 

Then there is the steepness of the slope.  If its too steep, only extreme skiers will chance it but if its too flat, you won't do much on it.  The ideal slope is 15 to 25 degrees.  In addition, they keep track of vertical drop or the distance from the top to the bottom because usually the larger the vertical drop, the longer  the ski run if its within the proper steepness. 

When planning a ski resort, the designers have to plan for carrying capacity which in this case refers to the vertical transport system.  They need to plan enough lifts to move the skiers up the hill so they can ski down.  It is vertical transport in feet per day divided by vertical demand.  Vertical transport in feet per day is found by multiplying vertical rise of the lift, by the hourly capacity of the lift by the number of number of hours in operation each day while vertical demand is the typical number of round trip runs made by a skier on one of the lifts.

If the resort has a ski rental place, the workers have to make sure the edges of the skies are set between one and three degrees. In addition angles are used when making turns, too sharp and you mess up, too shallow and you don't make the turn.  Furthermore, its good to know the turning radius which is marked on the ski.  It lets the skier know how tight a turn they can make.

Imagine assigning a project where students research what it takes to build a ski resort before creating their own.  There are a couple places on the web which have projects already set up for this.  One is at Teach Hub and the other looks at slopes.   Years ago I took a physics of avalanches class which was great because it included an assignment where we had to figure out where to put barriers so resorts or houses would not be crushed during an avalanche.  That was fun.

Warm-up

Warm-up

Providing Timely Feedback.

  I will be the first to admit that I struggle with ways to provide timely feedback.  I teach 6 different math classes, one of which is combined Algebra II and College Prep while another is called Fundamentals of Math made up of three groups all at different levels.

I finally realized students who asked "Is this right" provided a perfect opportunity for immediate feedback.  On the other hand, I don't always have the time to provide feedback as soon as I want. 

Feedback is much more than grading assignments and returning them.  A 95% indicates they know the material while a 50% tells you they know less but it doesn't tell you what they don't understand yet unless you analyze the missed problems.

it is important to provide task specific feedback where the teacher comments on things to improve.  I've been commenting that two or three words does not provide enough information when answering a question asking them to explain something.  In addition, it is good to ask students to provide self reflection on their understanding of the activity or material so the teacher can immediately adjust their teaching.

There are five steps in providing effective feedback to students regardless of the subject.

1.  It has to be immediate before they've learned it wrong.  One way is to post the answers so students can check to see if they did the problems correctly.  Another way is to utilize technology that provides that immediate feedback.  The goal here is to make sure students know they have not gotten it right before doing it wrong has become habit.

2. Feedback must target the students' specific need so feedback should be personalized and not applied to the whole class in a generic manner.  There are numerous sites which allow students to work a few problems and the site either provides the student with targeted feedback or gives the teacher a printout with information that can be used to help the student.

3. Feedback should be concrete such as when I noticed one of my students forgot to carry during multiplication.  Once I noticed that, he began to watch for that and began getting the problems right.

4. Do not just point out the errors, give them tools to help do the work.  Help students build upon previous successes so they continue to improve.  Remember a teacher cannot teach the material one day and expect them to take work home to do independently.  They will not have had time to learn.  Spend time helping them understand the concepts before expecting them to work outside of school.

5. Allow some of the feedback to help build their confidence because the more confident they are in math, the more likely they are to persevere and work their way through harder problems.

This is just a start.  I'll revisit the topic in the future.  Let me know what you think.  I'd love to hear.  Have a great day.

360 degree photos and Math.

  If you follow any crowd funding places, especially the ones with cutting edge technology, you'll have seen several offering fish eye lenses for various phones. I was recently offered a fish eye lens to pop on my iPhone to make 360 degree pictures and videos.  I don't own an iPhone so it makes no sense to buy it.

I've been trying to figure out how to incorporate 360 degree material into my class but its taken me a bit.  However, I  have found a few things. 360 degree videos are often referred to as spherical videos which use Mobius transformations for purposes of editing.

which are used to map a one to one mapping from one domain to another domain. 

Mobius transformations are used to map a one to one from on domain to another.  In addition, these are transformations of the sphere such as making regular rotations of the sphere, zoom like transformations, and other similar effects.  The first transformation turns the pixel coordinates into angles, the second transformation takes it from the equirectangular projection while the third deals with complex numbers.

On the other hand if you look at a 360 degree photo also known as a fish eye projection, it appears distorted but its not. Its actually a three dimensional projection onto a two dimensional plane.  This is what gives it the peculiar look.  There are programs which convert 360 degree photos into landscape shots so they look more "normal."

In simpler terms, the math involved takes an image which is circular in shape and creates a more rectangular shape through the use of "uncurling" the lines. Think of it this way, your source image is  2l by 2l and you want to make it so its destination image is 4l by 1l. Only the pixels inside the inscribed circle of the source make it to the destination image.

The pixels along the top of the destination image come from the circumference of the source image. The formula for the math conversion is (4l-x)/4l * 2pi where x is the Cartesian X axis.  In addition, when iterating from left to right on the destination image is the same as going clockwise on the source image. 

The radius (remember a circle has a radius) is calculated as l - y where y is the Cartesian Y axis.  So iterating from top to bottom on the destination image is the same as going from the edge to the center of the source image.

There are software programs out there with the math already present so you can convert your photos without having to do the math but its nice to know what these programs do when the conversion is carried out.

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

Persistence is Stamina

  As I looked up ways to increase student persistence, it came up as increasing student stamina.  I like that idea.  Helping students develop persistence is not just a goal in mathematics, it is needed for all subjects.

The English teacher comments on how students believe that one draft is enough to produce a wonderful award winning epic while the social studies teacher noted that once students get slightly behind, they feel as if they cannot catch up and give up.

There are ways to help students build  persistence or stamina.  Students need  persistence or stamina so they can work through challenges, deal with failures, and meet all goals they set for themselves. Unfortunately, most refuse to believe that learning math is like learning a sport. 

 One of the first things is to help students change their mindset from can't to can.  Too often, they have a little voice in their head which convinces them they cannot do it. That little voice tends to override the teacher saying they can learn.   One way is to praise them when they are focused on meeting specific goals.

Next, give a gentle push when they run into a bump by using encouraging words.  Its ok to let students know they can take a bit of a break before resuming work. Many students just give up rather than going back and many math teachers want students to finish it all in one sitting rather than acknowledging the brain needs breaks to function at peak efficiency.

Furthermore, it helps if the teacher models persistence by sharing an incident where the teacher overcame something.  It could be something as simple as replacing the toilet that broke but explaining the problems they faced.  It might be when they were in college and struggled through a class.

In addition, it is good to give students optional ways to talk to themselves so rather than saying "Its too hard." the could say, "I could ask the teacher for help."  or "It'll get easier with a bit more practice."  Most students tell themselves they can't do it rather than encouraging themselves.

The teacher also needs to hold students to high expectations while letting them know they can meet those expectations and providing them with the tools they need to do that.  Its also important to incorporate technology because it can provide immediate feedback.

One thing teachers tend to avoid is taking the time to explain how the brain learns and how it changes as it is used. Finally, incorporate repetition of these strategies so students become comfortable with them and increase their stamina.

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

Vectors in Real Life

  I usually teach vectors about once every other year and usually when I have to teach a semester or two of physics because its easy to integrate vectors there but easier than in Math.  Vectors are one of those topics that are much harder to teach without a context.

I had to do a bit of research so I can teach it in my College Prep math class.  I needed to know more about their use in the real world because its important to show relationships and connections.

Many of the ways vectors are used in real life are done so without using the word vector.  This is a good thing to shae with students. So on to the way vectors are used in real life.

1.  Air traffic controllers give pilots a specific heading (direction) with a specific distance (magnitude) along the planned route.Vectors are also used in flight patterns to take into account the wind blowing with or against or cross the plane's flight.

2.Vectors are used to aim cannons and other heavy artillery.

3. In baseball, any player must run in a certain direction going a certain distance in order to intercept a ball.  The player also has to predict where the ball will be to catch it, not aim for where it is now.

4. In basketball, football and golf, the player uses vectors to determine the trajectory of the ball in order to make a basket, get it to the right player to get it to the goal or in the hole.  In football, the quarterback must take his own movement into account, the receiver's movement and the path of ball, all of which can be represented by vector arrows.

5.  When someone asks for directions, they are given them via vectors such as two blocks left, then three blocks right.

6.  Many video games such as Angry Birds use vectors in the shooting of birds at objects.  The vectors include the amount of the pull used to launch the bird and  the angle of launch.  This is actually the most relevant use of vectors for my students.  Vectors are used in video game development to describe the location of an objects or physical simulations of objects.

There you have it, six different situations where vectors are used.  No one thinks of arrows or directions with speed but they use it every day instinctively.  If people want they can create a diagram of a football play off of the television or a shot from a basketball game but we know what we are doing without knowing all the proper mathematical terms.

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

6.

Writing in Mathematics.

  We hear more and more that students need to have the ability to explain what their thinking is but writing can play a much more important part in the math classroom.

Most math teachers are not trained in writing.  We are trained in mathematical formulas and solving for unknowns.  Thus when we have to integrate writing, we try or ask the English department but they don't have a reference to writing in mathematics.

It appears there is a minimum of two types of writing.  First is writing to learn which uses short or informal writing tasks designed to help students think about key concepts and ideas.  These activities are sprinkled throughout the lesson and focus on the concept, not on proper writing techniques.  This type of writing is found in journal writing, logs, written responses, etc.

The other type of writing is referred to as writing to demonstrate knowledge in which they show what they've learned and show their understanding of concepts and ideas. This type of writing requires students to write for a specific audience using more formal language and are checked for grammer, punctuation, etc.

To turn writing into a learning experience, it should include more than just copying notes.  It should personalize the writing by asking them to reflect, ask questions, which helps them better understand the concept. 

It is best not to assume students know how to write in mathematics. There are suggestions the teacher can implement to help students learn to write for mathematics. 

1.  Explain the strategy and its purpose.  Include real life examples if possible and who the audience is.  This gives students more understanding they are not only writing because the teacher said so. 

2. Model samples of the type of writing you ask students to do.  it is important to include the type of thinking that goes into the process both before and during. 

3.  Give students a chance to practice the strategy or process in small groups before having them do it alone.

4. Provide timely feedback and have students use the feedback when they rewrite the piece. 

5. Encourage them to become more independent.

As far as strategies go, most math teachers lack knowledge of strategies, especially if they are not trained in writing or work with students who may lack mathematical vocabulary. I found this 33 page write up filled with possible strategies.  Each strategy is described with information on how it helps the student, implementation suggestions, and examples. The pfd includes 14 writing to learn strategies and three writing to demonstrate knowledge. 

Some of these strategies I've seen before but most of them I haven't so I will have several more strategies to incorporate into my teaching, especially with my lowest math group, many of whom read and write poorly.

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

Warm-up

Warm-up

Making Connections.

  We all know how important it is for students to make connections to prior knowledge, to life and the world around them.  The other day, I realized I have to teach my students to learn to connect seemingly different things. 

The idea came from two sources.  The first is a book I'm reading called "Writing on the classroom wall" by Steve Wynborny who also does those Splat activities.  He said "Learning is about making connections." The other is my mother who talks about relatives and always includes the information of how they are related.  Always.

So I realized I needed to start getting my students thinking in terms of connections but that isn't easy because they've never had to think much in math till they got to my class.

I started this past Monday.  I wrote Steve's sentence on the board and explained we were going to spend the week tackling this.  So I used the warm-up for introducing this.  I love that it causes them to really think about what they are doing and I suspect they are doing some higher level, critical thinking.

This past Monday, I asked them to find the connection between two drawings.  They gave the surface answers like "It has a bunch of lines."  or  "They are both purple.".  I told them that is not a connection, that is a physical description.  After letting them struggle for a bit, I asked questions and got them to the solution of they are connected because of multiplication. 

The next day I asked them to explain how the moon and the oceans are connected. Many  of them googled it before writing down a complex answer on gravitational pull, etc.  Some discussed the size, some said I don't know.  I told them they are over thinking it.  One simple connection is all they need. Finally someone called out "tides".  I pumped my fist.  That was the break through point.  They started to look beyond the surface.  Some even started looking deeper.

Another time, I asked for a connection between music and cooking.  In addition to fractions, I got things like improvisation.  In jazz and in cooking you create your own "recipes" for the final product.  You have the guidelines to do it but you adjust and create something new within that frame work.

I had a few things like the tower of pisa and a can of coke.  They took to the internet to look up pictures so they could compare the two.  The universal conclusion was they are connected by their cylindrical shape.  Another was a clown and Rudolph.  Most everyone said the red nose but one girl said she was looking for a different connection.

This has been a great week doing it because my students are talking more, checking out the internet, and looking for those connections.  The next step is to start asking for connections in regard to the math I teach.  This is going to be much harder but it will happen if I take them through it a step at a time. 

It is going well and I like this better than my usual warm-ups because they are more involved and really thinking.  Let me know what you think.

Teaching Persistance

  Too many of my students give up when they think the work is too hard, especially when I ask them to work independently.  It is frustrating when I ask them to do one step and they shut down because they tell me "Its too hard."  If I do it all on the board, they come to rely on me too much and they never learn to do it.  In addition, they do not learn persistence.

I just finished reading an article on how the Japanese teach problem solving and persistence to students.  I'm  sharing it because the method involves more from students than the method most teachers here use.

The method is referred to as "Teaching through problem solving" not "Teaching problem solving."  The second is where we teach students the standard steps used to solve problems, especially word problems.  But teaching through problem solving is different.  In this method, the teacher sets up the context and introduces the problem before allowing about 10 minutes for students to work on it.  During that time, the teacher walks around, monitoring progress and noting which approaches are being used.

It is only after this exploration time, the teacher begins a whole class discussion to allow students a chance to share their ideas for solving the problem.  Rather than stopping here, students are asked to think about and compare the ideas, decide which ones are incorrect and why, which ideas are similar, which are more elegant or which ones are effective.  So they are having an in-depth discussion on the different approaches.

The idea behind this method is that students learn new material, ideas, or procedures using discussion. An example of how this plays out is as follows.

Textbooks are closed and the board is totally empty.  The teacher either projects or places a poster with empty rabbit cages of different sizes.  The teacher leads the class questioning them about their observations of the cages and certain assumptions they could make from what they saw.  Next the teacher displays each cage with rabbits for students to share their observations.  This leads to a student asking about crowdedness in the cages. 

The teacher passes out pictures of the cages with rabbits for students to glue in their notebooks. They refer to the pictures as they work independently to determine crowdedness.   As the teacher wanders around the classroom, observing work, he or she may ask what the student is doing and suggests they write the idea down in their notebooks.

After 5 or 10 minutes, the teacher asks students to share their ideas with everyone. The students write down the common ones, then continue pursuing their thought for a few more minutes before sharing more ideas. The teacher guides the discussion  through the main ideas and context till students have arrived at the actual lesson material.

I'd love to do this type of lesson but I know it will be hard because my students arrive in high school with a learned helplessness that I have to overcome.  It is all one step at a time.

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

Teach Word Problems as Mysteries.

  To this day, I have issues with word problems but I do try to teach them.  I often save the shortest day of instruction for activities such as math related art, math related music, or learning to solve the more complex word problems.

I've taught them using reading clues such as who, what, where, when, and how.  You know, who did it, what did they do, where did it happen, when did it happen and how was it done.  The last thing was the what do you have to find question.

I've also taught it as KFCW or what do you know, what do you have to find, what do you have to consider or thing about and then the work.  Honestly, the reading way worked a bit better but they still fight me on this topic.  Even many of the performance tasks on the internet are not anymore exciting or real than the ones in the text book.

This is one reason, I want to try teaching word problems as mysteries.  Lets look at the process if you treat word problems as mysteries. 

First, hook them.  Maybe have a mysterious envelope arrive, perhaps delivered by another teacher or the counselor, or the security guy. In the envelope is the problem written down as if it were being told by another person, rather than handing out a paper with the problem.  Or you could have a friend help create a trailer length video which sets it all up.

Second, take time to discuss the possible story behind the problem.  If the problem deals with cans of paint, why would you be ordering cans of paint?  Or if you are traveling across the country, why might you need to keep track of total mileage?   Nothing happens in isolation even though most word problems in the textbook do.

Third, take time to decide what you know from the "crime scene".  What do you have to find.  Sometimes, students have to find one item before they can find the actual information.  CSI technicians always write down every piece of evidence they find.  They draw pictures of the crime scene so students should draw a picture of the scene.

Fourth, have students take time to picture themselves in the situation.  Is there anything they have experience with in their lives they could call upon to solve the problem with?  Do they have prior knowledge that would be applicable?  If there a formula you can think of that might work?  If they can't think of anyone, maybe they could report back to someone to explain the problem, much the same as a detective who explains the crime to their boss.

Fifth, once the problem has been solved, have students ask themselves if this answer is reasonable.  They should also write out an explanation of how they arrived at the answer and check to see that the units are correct.

To help get the students into the right mind set, I purchased One Minute Mysteries: 65 Short Mysteries you solve with math from Amazon because I want to create a bit more interest and enjoyment in solving word problem mysteries.  I'll let you know how it goes in another entry later on.

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

Issues With Teaching Problem Solving.

  Unfortunately, problem solving in the past has often been ignored in favor of having students learn algorithms and processes needed to solve equations.  If problem solving was taught, it was taught using a standard technique such as identify key words, underline the important information, choose one method, and solve.

There are several reasons why problem solving is not taught in the lower grades in a way that helps students become comfortable with the experience.

First, many teachers are uncomfortable teaching problem solving skills because they do not have the skills themselves, or they are not comfortable with mathematics in the first place.  Some teachers believe they need to take a class on teaching problem solving because they have not learned it.  In addition, teachers often do not want to admit they don't know if a student's explanation is valid and rather than saying something like "Let me check it out and get back to you.", they don't teach it.

Furthermore, students are often uncomfortable with open ended questions because they are used to primarily solving equations with one answer rather than multiple possibilities. Their insecurity can cause them to shut down and refuse to do the problem.

Sometimes teachers feel that teaching problem solving skills take too much time away from covering the material in the curriculum so they ignore it. What they ignore is that many problems are designed to have students recognize patterns which translate to mathematics in general.  Once students are able to see patterns, it speeds up their learning.  Problem solving also allows students to take ownership and acquire greater understanding of the material.

In addition, it is often felt that lower ability students, students who struggle with reading or mathematics should not attempt problem solving but this is incorrect. Since all students should read the problems more than once, it can improve reading ability, and if the teacher has students restate the problem in their own words, it is easy to see if they understood it.

One way to help students with low reading abilities is to have the whole class read the problem together.  If the teacher prepares an audio version of the problem, students can play it and listen to it as many times as needed.

Unfortunately, most problems that require real problem solving such as word problems, tend to use exactly the same methodology as the equations in the section and most word problems have a feel of having been made up to fit the section topic.  In addition, many of the word problems are not complex, even in high school. 

I sometimes think, solving word problems should be compared to solving mysteries.  Let me know what you think, I'd love to hear. 

The Advantages of Using Mathematical Riddles.

Mathematical riddles are something we seldom see in class because its not something most teachers know about. 

Mathematical riddles are not usually found in a familiar wording.  I remember one riddle from my childhood. "What is black and white and read all over."  Answer " A newspaper." 

Most mathematical riddles are like the one from Sunday's warm-up.

Mathematical riddles are actually more like logic problems which require mathematics to solve.  So why should mathematical riddles be used in the classroom.

One good reason for using mathematical riddles in the classroom is that they strengthen both reading and problem solving skills.  In addition, they encourage critical thinking skills and can provide motivation by making math fun.

Another important feature of mathematical riddles and problem solving is that it helps encourage mathematical development based on current knowledge.  In other words, they can use the skills they know to solve a problem even if the riddle is designed for a higher level of mathematics.  Furthermore, the challenge of solving riddles can make mathematics enjoyable for students because they are presented in a different way then standard practice problems.

When students have to struggle with problems, they acquire a deeper understanding of the mathematics involved and understanding is enhanced.  In addition, they way they choose to tackle a problem is like a scientist approaches a research problem so they get a better feeling for the way mathematics works.

Furthermore, riddles allow students to explore ideas while extending their creativity in solving the problems.  When students are allowed to collaborate when solving riddles, they are offered the opportunity to verbalize ideas which helps clarify their understanding. Group work increases enjoyment, learning, and allows social skills a chance to improve as they communicate with each other.

The other great thing about math riddles is that the problem solving skills they learn can be applied in a non-mathematical situation because many of the skills used are general.    It is important that other subjects teach students to apply these same skills so they understand that problem solving is not confined to Mathematics.

A simple search for math riddles will bring up quite a few sites with problems designed for all ages.  Let me know what you think.

Warm up

Warm-up

The Math of Fitness!

  Fitness is a huge industry in today's society.  Look at how many gyms have sprung up, how many are into providing 24 hour service, how many other places are offering fitness.  It is a big business.

There is math in BMI (body mass index), heart rates, time needed to burn calories, loosing weight, etc.  The good thing, is that much of the math used by fitness professionals includes percentages and decimals.

Here are some examples one can share with students who want to know "When will I ever use this?"

1.  Changing percents to decimals are used to calculate the percent of predicted maximum heart rate, finding the ideal weight of a person with 30% fat, determining the percent of fat when given the amount in grams.

2. They need to use decimals to figure out what a persons body weight should be if they have a certain percent of fat. 

3. BMI provides an estimate of body fat based on height and weight. The formula for BMI is your weight in kg/height in meters.

4.  There are a variety of body fat calculations that can be done.  Body fat calculations means if you have a body fat of 25%, that means 25% of your total body weight is the amount of fat you carry.  The lean body weight is the body weight minus the fat weight.  So if you weigh 100 pounds with a 25% body fat rate, 25 pounds of that weight is made up of fat while 75 pounds is the amount that is your lean body weight.

5. Another calculation is the BMR or Basal Metabolic Rate, the number of calories needed by a person at rest.  The general equation is different for men or women, but its basically a starting number + a factor times the weight + a factor times the height - a factor times the age.  Then the BMR is multiplied by another factor to determine the number of calories needed to maintain a persons weight.

6. There are all sorts of calculations in regard to heart rate.  There is the at rest heart rate, the maximum heart rate and heart rate reserve which is the difference between the resting heart rate and the maximum heart rate.  Add to that another formula to find a target heart rate which uses the heart rate reserve divided by the percent intensity added to the the resting heart rate.

7. There are also equations associated with calories, gaining weight, loosing weight or your total caloric expenditure.  Then one has to take into account there are 3500 calories in a pound so this is good information if you want to loose or gain or even maintain your weight.

The above equations are commonly used by everyone who is into fitness.  The individual equations are easy to find and use.  It shows ow math plays an important part in fitness.  Without it, they couldn't do much. 

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

One Step Equations + Real World

  When I learned to solve equations, we solved them without any real world context.  It wasn't important.  When we did have "real world" problems, the books were so old that we'd see questions about Roman ships and the number of slaves needed to row it.

I've noticed that my students have issues with writing equations for real world situations due to having learned to solve one step equations in isolation. 

In a couple of weeks, my Pre-Algebra class will be starting the topic.  Instead of teaching it the usual way in isolation, I'm going to begin with real world situations such as buying soda's or pizzas so they can call on previous knowledge while being able to relate it in context.  I think its a shame we wait until students have learned the mechanics before most of us introduce the real world contexts.

As far as the mechanics, there are tons of activities out there which allow students lots of practice.  One site has suggestions including math mazes.  I've never heard of a math maze but it makes practice a bit more interesting. If you haven't seen them, they are a bit like flow charts with the problem and arrows with possible answers.  If you choose the correct answers, you'll get through the maze to the end.  If not, you don't so you know if your answers are correct.  Another mechanical way is to provide tic tac toe  activity where two students play it the normal way, except they have to solve the equation before they can claim it as a O or X.

But most of the material I've seen seldom integrates real world applications with the standard practice worksheets and activities.  I am going to introduce the topic by asking them to write a word problem to go with 3x = $59.85 or x + 5 = 6.25.  I like the open questions because they allow for multiple correct answers.

The next step is to provide a word problem which they have to turn into a one step equations so there is  the connection.  I also plan to weave videos through this part to reinforce learning and help them work on their active learning skills.

Throughout the unit, I want students to create word problems or situations to go with solving one step equations.  I'll throw in the math mazes, tic-tac-toe games,  and the ever wonderful snow ball fight, where students write a one step equation on a piece of paper before crumpling it up and throwing it around until the teacher says stop.  They open up the paper, solve the equation, then crumple the paper up again and throw it until until the teacher says stop.

Students need to practice the mechanics but they need to do it in a more fun way so they don't feel as if they are just answering problem after problem for no reason.  If students see no reason to learn something, they won't be as motivated as they might be.  I'm hoping by connecting it to real life situations, they will have a better understanding.

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

Brain Friendly Learning in Math

  Yesterday, I discussed associated learning in general. Today, I'm going to look at it with specific applications to math.  I do this because I've had people discuss wonderful topics and techniques but I had to figure out how to use them in mathematics. The presenter had no idea.

Remember that math builds on itself so its important to create a situation so the brain itself moves the information from short term to long term memory. 

In mathematics or any topic, teachers can use a seven step process to help brains move from passive to active learning.

Step one is to reach students because if you don't reach them, you can't teach them.  One way to do this is to create a question or hook of some sort to grab their attention.  The hook might be a movie trailer designed to spark their attention. 

Step two is to give students an opportunity to reflect on the material so their brains can begin to make connections between prior knowledge and current material.  This can be done through a journal entry, a reflective question, or having two students discuss it.  Nothing wrong with using journal entries to incorporate writing into the math classroom because it helps develop student ability to express their thinking.

Step three is to have students put the ideas into their own words.  Have students write down a short summary of what's been discussed.  Teachers need to make sure they have the correct information because its at this point that the material is still in the temporary memory and must be correct before it moves to long term memory. This is important because students need practice in expressing mathematical ideas in their own words, especially if they are English Language Learners.

Step four is to reinforce the learning.   As students write down their understanding of the material, it is important for the teacher to provide immediate feedback either through positive reinforcement if correct or providing correction if its not correct.  Everything I've read indicates immediate feedback is important.  If the misconception reaches long term memory, its difficult to correct it so this is extremely important.

Step five is the stage where differentiation happens.  If students need reinforcement, this is where the teacher provides a new hook or reflection while those who have a better grasp will study the topic in more depth or practice it more.  If the topic is brand new, you might have additional rehearsals

Step six is the review stage where students play games, write, draw, create mind maps, or perhaps include a practice test to see where they are in understanding the material.  I use Jeopardy, Kahoot, play videos with music on the topic.

Step seven is when students show if they can retrieve the material.  The more practice students have, the less stressful this step is.  This can be done through a test or by having students create the assessment they thing would show what they know.

All of these steps are easy to incorporate into the math classroom. I always like things like this that are done in a step by step manner so I can incorporate it immediately rather than figuring out how to include it.

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

Accelerated Learning.

No this column is not about that math and reading system offered to schools.  I am referring to a topic I'm seeing in educational books.  Recently, I've seen books on Accelerated Learning or Brain friendly learning.

This type of learning refers to any activity that expedites the learning process.  The idea is to create a situation where people learn the material faster while retaining it for longer periods of time. 

This is not the same as cramming.  Something most people has done at some time in their life.  I did it only in so far as doing a review of all the material on the test but it was still trying to cram.  Accelerated learning has five stages to the process.

The first stage is to prepare both the mind and environment.  It is recommended a person find a nice quiet place where they need to sit upright.  Laying down is not good for studying because it makes people more tired while decreasing concentration.  Always look over the material and set a goal time wise to learn the material. 

One way to do this is to imagine yourself as the instructor of the class.  As you study the material decide what you want to teach in what order so you can picture how you'd teach to others.  By doing this, you are internalizing the material, making it easier for you to learn it.

The second stage involves using the whole brain.  The left side of the brain is the part of the brain that produces language, words, and numbers while the right side of the brain is the artistic center responsible for music, patterns, spacial relations, etc. The final part, the limbic system decides if the information is important enough to remember.

There is research indicating that music without words with a tempo of about one beat per second. The music helps relax the brain while stimulating the limbic system.  It is important to take notes while reading or watching something but it is just as important to stop and picture how the material fits together overall. 

In addition, stop and repeat material out loud to reinforce your memory.  Furthermore, always rewrite your notes to add that last bit of learning.  Be sure to include short 5 minute breaks every 30 minutes so your brain has a chance to process information.

Stage three is to understand the material and not just know it.  It is possible to know something such as a quote from Shakespeare but to understand the quote is a whole different level.  To effectively learn, we need both understanding and knowing and must do it by turning passive memory into active memory.

The most effective way of doing this is either by explaining it to another person, organizing it, or summarizing it.  For instance, if you take notes, always paraphrase them by putting them into your own words.  Paraphrasing material means you have to think more deeply about the material while developing a better understanding.  In addition, if you organize the material from most important to least, you are thinking deeply about the topic.  If you can make the material your own, you will remember it better.

One way to summarize the information is to create a mind map with the main topic in the center.  Reaching out from the center, place keywords which describe important ideas and connect to the center.  Continue this activity until all the material is covered. Keywords trigger information about the topic.  If the mind map is done with color, pictures, shapes, symbols, etc, it makes it easier to remember the material.A good way to remember the material is to try to recreate the mind map on a different sheet of paper.

Another way to summarize information is to write one sentence on a paper that summarizes a paragraph.  Do this until you've finished the chapter.  Then take and summarize each page or half page with one sentence so you have a compact summary.

Stage four requires reviewing the material because most people do not remember material unless they look at it several times.  Begin by summarizing the material at the end of the study period, then the next day for several days, then once a week, once a month and finally after about 6 months, you should know the material well.  To make this effective, try to remember the material from memory rather than just rereading it.  One way is to pull out an empty paper and write down everything you can remember or recreate a mind map.  The review should not take more than 2 or 3 minutes.

A good way to remember the material is to associate pictures or some form of visualization since we remember pictures well.  This helps us create an association between the abstract and concrete which our brains need. 

Stage five has you apply the material.  There is not reason to learn the material if you have no where to apply it. So it is necessary to figure out ways to apply the material.

Tomorrow, I'm going to focus on applying accelerated learning techniques to math.  Let me know what you think I'd love to hear. 

Happy 2018