Mathematical Bingo

The other day, I wanted to run a bingo game using math equations and verbal descriptions.  That type of game is not always easy to find on the internet so I figured a way to do it myself.  It wasn't all that hard to set up.

First thing I did was find a bingo card templet on the internet.  One that allowed students to write in their choices.  I made copies for people so they could fill in their choices from a huge list on the front board.

I found a wonderful random name drawing generator at Name Picking Ninja because it allowed me to input a list of terms and I could eliminate the answer and not use it again.

It took students about 10 minutes to select 25 out of 40 algebraic expressions on the board.  I let them customize it so they'd have more buy in. 

The add names is where I added my list of predone choices I typed up in word.  I had to delete a list of writers but that took a short time.

When ready, I hit the go button, the choices spun and it stopped on one as you see in the photo.  I read the verbal phrase out loud.  For some of the classes who struggled more with this, I asked what product meant or second power to help them talk about the math more. 

The first one who managed five in a row got a small piece of candy as a reward.  Overall, it went well because it helped encourage conversation and it gave students practice in connecting the verbal with the algebraic expressions.  The down thing is that it was a bit slower since it involved discussion.

I will be able to use this in other classes to just as easily and I can use vocabulary, slope, linear equations, etc to test their knowledge.  I chose this venue so when they have a quiz on Friday, they might do better since we actually used it in a situation.

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

Complex Probability

I love simple probability but complex is a bit harder to do because many students have difficulty with the idea of recalculating probability for each case.  I admit when I was in high school and had the topic, I wondered why if you had two bags and selected from one bag why you needed to worry about the one you weren't using.  It wasn't until college I realized one of the probabilities was choosing one of two bags.

My high school teacher didn't explain much in detail because he thought it should be obvious to us as we looked at the situation.  I always hated the "Its Obvious" train of thought.

One nice way to introduce the idea of complex probability is to use the video Menu Tossup which looks at calculating the total number of combinations on the menu.  Something students can easily relate to since they like eating at any of those fast food places.

Another way to show more about complex probability is to set up a probability bingo game.  The math equals love site has a great explanation of the game and where it originated. The idea is to fill out a bingo card covered with various combinations such as blue/green, green/red, red/blue, green/green, blue/blue, or red/red.  Students tend to prefer playing games to completing worksheets.

In addition, you could create or purchase compound probability task cards which can be placed around the room so students can get movement while working their way through the activity.  You could also create QR codes with compound probability questions or create or buy a compound probability scavenger hunt.

Scholastic has a nice little unit on compound probability with everything needed to teach the lesson including the 5 accompanying worksheets.  Its geared for 6th and 7th grades but it could easily be used in the high school, especially if working with ELL students.  It has students use a tree diagram to help derive the formula.

On the other hand, Better Lesson a nice lesson for 7th graders to learn more about complex probability and sample space.  In addition, it has students determine probability using an arithmetic strategy.

Unfortunately, many times including probability and statistics in the Algebra, Pre-Algebra, or Geometry classes is a bit hard because they do not fit nicely into the flow of the classes.  The topic is almost a disruption in the flow but I save it for short weeks like Thanksgiving, the end of the semester, spring break where a natural break falls in the class.

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

How do you teach it?????

At some point we have to teach either probability or statistics either as a part of our regular math classes or as a stand alone class.  Not many of the books I use have really interesting lessons to use other than the standard penny flips and possibly the rock, scissors, paper game. 

I'm always open for some ideas to use in my classroom.  I've found a few and its time to share them with you. 

Fortunately, the American Statistical Association has some lovely peer reviewed lessons for teaching statistics.  The lesson are designed for grades K to 12 so if you have students who are not up to grade level, you can choose something designed for younger students or take a lesson and adjust it upwards. 

For middle school, a couple of lessons really caught my interest.  The first is "How random is the iPod's shuffle" which many students easily relate to because they use the shuffle button.  The other is "How long is 30 seconds?" which appeals to me because most students I know who try to count 30 seconds tend to rush it so they finish about five seconds ahead of the time.

High School has a lesson "Confidence in Salaries in Petroleum Engineering" examines the salaries in a field which is said to have one of the highest starting salaries for any profession.  Another one, "10,000 steps" examines one of the newest fitness trend which states you need 10,000 steps per day to stay healthy.

These are not one page lessons but more like 15 to 30 pages depending on the lesson.  They come complete with the objective, standards, materials, student sheets, etc. Everything needed to carry out the lessons properly.

Most of us have to introduce students to probability in a way they don't get bored or allows them to get up and move around.  What about setting up a QR game with 16 QR codes spread around the room and each QR code asks a probability question.  The question might be something like "You flipped two coins, what are the four possible answers?"  The student then writes the answer down on the answer sheet.  If you don't have the digital device, you could create the 16 questions and post them around the room.

Shodor.org has a really nice interactive spinner which the student to change the number of segments so students change check out the probabilities for different scenario's.  In addition, there are activities on fair chances, coin toss, theoretical vs experimental probability, and random number generators.  There are also worksheets and discussions. 

In addition, you could introduce probability using Bill Nye the science guy.  Episode S04E08 can be found on School Tube or You Tube.  Bill Nye is a fun person to watch as he explains an assortment of topics. 

The above probability activities are to teach simple probability but what about complex probability such as you have to remove two marbles without returning them to the mix.  How does that change things?  Tomorrow, we'll look at activities for complex probability.

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

How Do You Tell?

We all know that statistics can be in a positive or negative way and we've seen when they are not being properly used. Unfortunately, we can't always tell at a glance that the numbers presented are right. 

Often we hear when statistics are not being used properly, when two graphs are shown with different scales to convince us one product is better than another or one company is better than another. 

Yet there are other numbers out there that are used properly.  How do you help students understand which stats found in life are good and which can be ignored.  Part of teaching mathematics is to help them build a mathematical sense.  A sense of which things such as stats are reasonable and which are not.

If the statistics are used to promote something such as a solution to a problem or a course to help people, chances are the numbers provided may not be fully accurate because advertisers do not meet the same standards that a researcher does.  This means the "Three out of four dentists" recommend the protective power of this particular toothpaste might be suspect because the company wants you to buy it.

Second, if a statistic is used, can it be verified.  Where can I find the "Three out of four dentists" research showing this.  Any statistic that is worth anything can be verified because you should be able to find the study it is based on.  If I hear that some basketball player is able to shoot 4 out of 7 free throws, I should be able to go find that stat in a reputable site such as put out by the NBA.

Third, if the statistic does not seem to line up with reality, question it.  For instance, when I hear that three out of four marriages dissolve in divorce and out of everyone I know most have not divorced but there are a couple who may be on their third, fourth, or fifth marriage, I'd question it.  The multiple divorced people might change the statistic so it seems as if there are more than actually occur.

On the other hand, most sports teams tend to provide verifiable statistics on all players because these stats provide the owners with the necessary information for them to keep, trade, or release players.  Other groups tend to be insurance companies, actuarial companies, census bureaus, and other companies who use the finished data for setting rates.

Insurance companies have had a lot of data to come to the conclusion that females who reach 21 should pay less than their male counterparts since this is a standard practice across the board.  On the other hand, when I read the number of children killed has doubled every year since 1950, I can question that because its easily disproved by simple mathematical equations.

We need to help our students learn to distinguish good stats from bad stats and help them learn to tell the difference.  Let me know what you think, I'd love to hear.  Have a great day.

Doubling Pennies Etc.

  I researched the history of chess and one of the legends on its origin made me think of the doubled penny problem we often pose students in class.

The problem is the one where you ask a student if they want $1,000,000 or a penny doubled each day for a month.  After exploring the math, they discover the second is the better option.

The story I read said a wise man invented the game of chess to show the king how important each person in his kingdom was.  He enjoyed the game so much, he required all his subjects to learn the job and as a reward, he offered the wise man gold and silver but the man turned it down.  Instead, he requested one grain of wheat on the first square, two on the second, three on the third etc until all 64 squares had been used.  This would have given the wise man quite a lot of grain.  As a matter of fact, the story says the servants scrambled to find enough grain to do this.

Another place this particular doubling problem is found is in bacterial growth.  It is often expressed as starting with one that splits into two, each hour or day and you are asked to find the final numbers at the end of a specific period of time.  Its the same type of problem as the previous two.


All three problems fall under the general term of a doubling problem or exponential growth which is usually applied to how long it takes for a certain population to double.  One way to explain this to students is that it takes over one million microorganisms per milliliter to make water cloudy or contaminated so if it starts with one organism, it may not take long for the bacteria to split into one million organism.

So harmful bacteria grows at the same rate as good bacteria. If one harmful organism enters a pot of stew and splits every hour, the population would have grown to more than eight million which is enough to make someone sick.  This is one reason why you should not leave a pot of food in your trunk on a hot day.

I've seen the bacterial growth problems but I've never seen them applied to harmful bacteria in food which is a real life application of math and explains why there is an occasional outbreak of food poisoning at community potlucks.  It gives students a perspective on why you might want to care about bacterial growth.

So the next time I teach this topic, I'm going to include the example on harmful bacteria which people might find more interesting.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Warm-up

PEMDAS? Really?

Why do we continue insisting on teaching students to use the rigid rules of PEMDAS, also known as the Order of Operations?  Most students have trouble with this mnemonic because they think they should do all the multiplication, then division, then addition, then subtraction in that order.

The other issue is that people often get confused when there are multiple groupings within a problem because students sometimes forget to do all of the operations within the associated groupings.

I just read something I do not agree with fully but the author pointed out we should have students think of 8 - 2 + 1 as 8 + (-2) + 1 so they get the correct answer.  Rather than doing all the addition first rather than the addition or subtraction based on which is found first, second, etc so many see this problem as 8 - (2+1) or 8 - 5 = 3. 


In another article, it has been pointed out that parenthesis and group symbols are not actually operations but containers for the operations so its important for students to understand that the math needs to be done inside these groupings first following the proper order.  If there are grouping symbols inside of grouping symbols inside of grouping symbols, I was taught to begin inside the inner most grouping symbol but students often get confused with which grouping is the inner most one.

Furthermore, another artist pointed out that order of operations do not work the same if the problem is written in reverse Polish Notation which I have not used since I got rid of an HP calculator.  The same author stated we really need to quit teaching order of operations in this manner, rather help them develop the feel for solving numerical problems.

Another person classified PEMDAS as a computational strategy, a strategy and not a set of rules to be followed in stone.  I like thinking of it as a strategy rather than a set of rules because it encourages a bit more flexibility and less adherence to  a rigid application of rules.

One big thing I've noticed is that students arrive in high school having learned to do the order of operations incorrectly.  Sometimes I manage to correct their misconceptions but other times, I cannot change them and they continue applying the "rules" in the wrong way.  It would be nice if elementary teachers could teach this topic properly so students arrived knowing how to use these strategies properly.

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

Operational Vocabulary.

  Its the beginning of the school year and many teachers including myself begin with a review of vocabulary used to express various operations.  We ask students to change numerical sentences to written and vice versa.

Student success of this topic depends on two things.  First, students need the ability to understand the literal meaning of the sentence and they need the ability to write the same sentence mathematically.

It has been shown that students who struggle with reading often struggle with choosing the proper operation based on standard vocabulary.

One standard method used is to have students learn key words which works well many times but it is not always reliable.  One example is per.  The word per when used in this way - "Marty bought 25 pencils at $.15 per pencil" indicates multiplication while it means division when using it this way "Mary and her three friends shared 24 cookies.  How many cookies were there per person?" Yet, so many lessons only list per as multiplication or division.

I read one lesson plan which begins with passing out four index cards to each student to write four different math words which indicate an operation.  The teacher collects the words and writes them on the board.  Students have divided their pieces of paper into four quadrants with each quadrant labeled as addition, subtraction, multiplication, and division.  Students divide the words into four groups, each group representing an operation and write them on the paper, each word in its appropriate quadrant.

I think I'd take this a step further by having collecting the words, writing them on the board, have the students separate the words on a piece of paper but once all the students have completed this, ask them to walk around and check out other people's classifications.  After 5 or 10 minutes, it would be time to complete a class suggested four quadrant list of operations but each would have to provide justification for classifying the words. 

If a word has more than one possible operation such as "per", bring up context and how you would determine which operation should be done. What wording would provide a tip as to the operation indicated.

Once they have the words broken up into quadrants, give the students a word problem such as the difference of a number and eight is twelve and have them rewrite it into an addition problem such as twelve minus a number is eight.  I believe this will help students learn the relationships between addition and subtraction, multiplication and division.

If we teach students to use keywords to identify operation, we have to make sure they know which words could mean two different operations depending on context.  Too often teachers teach key words to represent only one operation rather than looking at the context of the keyword.  I'm getting ready to do this with one or two classes and I plan to bring up context.  I'll let you know how it goes. 

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

The Geometry of Hand-Sewing.

  Most days, I surf through Amazon looking for interesting books on an assortment of topics.  During my search I came across a book titled "The Geometry of Hand-Sewing" by Natalie Chanin on sale for $2.99 instead of the $18.00 digital price.  The title was enough to cause me to purchase it.

In the introduction, she stated she was introduced to geometry via a book "The Dot and the Line: A Romance in Lower Mathematics" by Norton Juster. 

She also mentions Euclid's "Elements" as the love letter to Geometry in which he explains how points and lines working together create all the shapes in our world. She also stated that she loves the relationships between points, lines, and surfaces which to me is such a profound statement since most geometric figures can be explained via the use of points, lines, and surfaces. 

Natalie goes on to explain that when she and her coworkers examined and compared various stitches, they were all based on the geometric grid system.  So the rest of the book after the introduction, is spent showing people how to apply the grid system to various stitches.  The grids might be square, diamond, rectangular, triangular or isolated circle. 

These grids provide the pattern for the stitching.  People are expected  to download and print the appropriate grid card with the correct grid and scale before transferring the information to the material in order to create the stitches. 

Before teaching a specific step, the author lets the reader know which grid is being used. For instance, the straight stitch which is simply a line where the needle comes up and goes down on equally spaced apart uses the one row square or diamond grid.

So each stitch or group of stitches are grouped together based on the grid they share in common.  Its amazing because they can create variety of stitches, including ones that are circular shaped from these grids and most are based on straight lines.

This books takes the basic tenants of geometry to create the stitches.  I love the book because it gives me a new way of presenting geometry in a way some of my students might relate to.  When I learned hand sewing, they never taught it using grids.  It was the watch, copy, and hope you learned to eyeball the distances to create the work. This is awesome.

Let me know what you think, I'd love to hear.  Have a great day and check out the book.

The Cost of Terraforming.

 






You’ve seen it appear in movies, television shows, or in books.  The scene usually shows someone launching a “seed” which causes a planet to transform from lifeless to fully capable of supporting life in a short period of time but when it’s used in the media, the cost is never discussed.  If it takes a short time toterraform, planets are no longer valuable.

When I speak of terraforming, I refer to finding a barren planet or exoplanet, create a survivable atmosphere, a soil in which one can grow plants and trees for food, housing or other items, water, rock, nitrogen, and carbon cycles,and everything else required for people who want to live there.

Sine it would take a long time to terraform a planet, there are various costs to examine.  First thing to consider is terraforming will require a long term investment either by a company or by the
government.  If a company is investing in this project, the company provides the funding and reaps the profits whereas if the government invests, they tend to look more at who in society benefits rather than looking at the profit.

If anyone looks at terraforming a planet, they must consider the cost of sending the materials out there.   The cost of sending the materials might be too expensive to ship so is it better to wait a number of years for improved technology which should lower transit costs which could be repeated for many years or do you take the money and use it to clean up the earth?

 How do companies fund such an item?  Through long term planning or via fundraising via crowdfunding.   The company may also look at funding it via multiple stages and each stage has to be looked at in terms of making a profit.

I had not realized there were so many costs to think of on this topic.  Let me know what you thin. I would love to hear

Another day of flight and fun

Still at conference

Asteroids and FEMA

Asteroids inhabit our Solar System in bands and in small areas.  Some have their own satellites or moons and so many follow a specific orbit.  Many are seen once but their orbit is so large they appear when a different generation is watching the skies.

Today, I attended a talk on asteroids where the first half of the talk was devoted to explaining all sorts of things associated with asteroids.  Of course, there is a ton of math involved in this whole topic because scientists want to know when an asteroid is likely to hit the earth.

This has lead to FEMA developing emergency plans should we be hit by an asteroid.  In addition, scientists are working on creating plans to move or destroy the asteroid.  If they change its course while it is passing near the earth so it cannot hit the earth, it could possibly hit it another time.  If they blow it up, there is all the rubble which could hit the earth or possibly assume a new orbit.  So now there are more orbits to keep track of.

There is lots of math involved in calculating orbits and orbital change so they can theoretically predict when paths will cross.  In addition, the are hoping they are putting in a rate of travel that is fairly accurate.  Apparently, one thing that happens is asteroids tend to crash into each other naturally breaking them up.

Scientists have been producing bar graphs of things such as sizes of the asteroids which hit the earth and the results have been used to calculate the probabilities of asteroids of certain sizes hitting the earth.  The chances of an asteroid that has a diameter of more than 1 km hitting the earth works out to us being safe for the next 250 years.

I wish you all could have heard the talk.  It was fascinating.  Let me know what you think, I would love to hear.

Rotating Space Station

I am attending a conference with lots of math, science, and writing.  I traveled all night and began attending presentations right away.  One presentation I attended was quite fascinating in that it looked at all the viewpoints in reference to rotating space stations.


The rotating space station was first theorized back in 1903 and scientists such a Braun looked at ways it might be built.  The gentleman used vectors in the form of vector arrows to explain why the movement occurred.

He showed the movement of the person inside the rotating station from both inside and from the space ship circling the station.  It was obvious the Coriolus effect effected the path of the person who was picking, running, falling, even playing basketball. One reason puking was covered is because it happens under certain circumstances.  In addition the path it follows is based on whether it is sent retrograde or prograde. Prograde is less messy.

So in addition to using vector arrows, he explained how the Coriolus effect applied to certain situations and why the viewer saw the movement the way they did.  At the end he gave a brief rundown of the equations these observations were based on.  Apparently, a rotating space station may not be the most stable building because the Coriolus effect could be quite wobbly.

The audience laughed so much because reality was so different from the expected reactions.  I found it interesting because it provided a real look at physics and math in a great situation. I will try to share more tomorrow from a different talk.

The First Days of Class

 As you know I work with ELL students in the wilds of Alaska.  On the very first day of school, we welcome students, go over things, and pass out permission slips etc for students because the next four days they will be out of the building doing something. 

Middle school students take daily trips out on the tundra to learn more about the plants and animals their ancestors ate and how to use them.  They might fish one day, play native games, or run around.  The high school goes out camping and does the same type of thing.

It isn't until the second week we actually start teaching and that is when I have certain activities I give the students every year.

1.  I pass out a textbook scavenger hunt for them to become acquainted with their textbooks.  They do not like to read and will work hard to avoid doing much with it so this is one way they will find the answers, index, glossary, and other important pieces of information.

2.  By Wednesday, I'll start students on learning to use Cornell notes in Math.  The science teacher and the social studies teachers also use this method in their classes so they will get a strong introduction to it. 

3.  After they've been reminded about Cornell Notes, I take time to show them how to outline the book using a straightforward outline but they will still do vocabulary and examples in their Cornell notes.

4.  Usually just before they get the scavenger hunt, they get the syllabus so they know what is happening and the 10% decrease per day they are late.

These are the four big ones but this is compounded by MAP testing about the same time.  Makes it hard to really do much teaching till the 3rd full week.

I think its important to help students learn skills which will carry through to any advanced training they choose to go to.  Half the struggle in my math classes is helping the students learn to look at their notes for review or for help when doing problems.

Most of my students arrive in high school with no study skills, no ability to read the textbook or take notes.  They even have difficulty putting ideas and definitions into their own words.  Unfortunately, they often think that if they find something on the internet, they can use it without acknowledgement or without paraphrasing.  All the high school teachers struggle with that last one.

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

NOTE:  My internet is finally up at home but I'm off to a conference beginning tomorrow so I'll try to share things I'm learning as it progresses.  If I don't post, know I'm learning things to share.

Extrovert Vs Introvert.

According to something I just read, it appears the brains of introverts and extroverts are rather different.  The big difference between the two is how they use the environment.  An extrovert is someone who gets energized by outside sources such as people, activities, or things while an introvert is energized by internal things such as ideas, emotions, or impressions.  Extrinsic vs intrinsic.  Extroverts need outside stimulation while introverts need inside stimulation.

Research discovered that the two groups process information differently while using different parts of the brain and different neurotransmitters.  Apparently, extroverts access information in the short term memory to develop thoughts while introverts locate the information in their long term memory used to build more complex thoughts. 

In addition, the brain releases dopamine also known as the "reward chemical" which extroverts experience more of due to their being stimulated by external items.  In brain scans, extroverts showed more activity in the areas associated with pleasure and more dopamine is released although both introverts and extroverts have the same amount of it.  Since introverts release less dopamine, they find the livelier situations more exhausting.

So as they think, they act differently.  The extrovert is more likely to participate in a lively discussion going on while the introvert appears to be uninterested and unwilling to participate when in fact they tend to be processing the information and do not think about volunteering.  Introverts prefer to work independently or in small groups while the extrovert can work in larger groups.  Introverts also have a reputation for being able to explain things better.


Consequently, extroverts find the social situations more rewarding and more often seek positive social rewards.  Since classrooms are filled with social rewards such as grades, parties, etc, extroverts find this perfect because it energizes them.  On the other side of the coin, giving a participation grade can hurt introverts because they do not "perform" well in front of the class and they often do not get full participation points.

It is possible the student who sits in the back and appears to be unmotivated is actually an introvert.  Remember an introvert is much more comfortable being allowed to observe but hates to be pressured into participating.  However, you can use a rotating system where you call on all students to respond but make sure to give them time to formulate an answer. Another thing is to give students the information so they can prepare questions, thoughts, etc for the next class. 

This is a short breakdown of the differences between extroverts and introverts.  I hope this helps you think of new ways to reach students who are often forgotten or penalized for being an introvert.  Let me know what you think, I'd love to hear.

Motivating the Introverted Student.

 There is always at least one student in your class who is so quiet you can easily forget they are there.  They don't participate in general conversations, they seldom ask for help, they try to be invisible.  They are the group who often fall by the wayside because of being so quiet.

There are things we can do to help these students do well in math and so they no longer fall through the cracks.  We sometimes have to work with them so they are more willing to participate.

Lets look at several things we can do as teachers to help them do better in class.  In general introverted students prefer to work one on one or in small groups, they prefer to think a while before sharing in the think pair share activity, take time making decisions, assessing risk before doing something, need quiet environment to recharge.

So what are some strategies to help these students?

1.  Allow students to write answers on a postit note before giving verbal answers.

2. Have a quiet corner for reflection and recharging.  Could be a nook or a corner.

3. Give students one minute to reflect on their learning by using a prompt.

4. Include  one on one activities.  One I like is to start with two people who talk, then add the two groups together for a group of four etc till you have a big group.

5. Count to 10 before calling on students.

6. Hang an image for student to look at for several minutes before they share their thoughts.

7.  Provide a preview for the lesson such as an essential question, an agenda, syllabus, daily schedule, or preview of the unit.

Tomorrow, I'm going to take time to explore the difference in how the brain works for introverts and extraverts.  Let me know what you think I'd love to hear.

Note:  I am still waiting for my internet at home.  The company changed billing systems and my account didn't flip over so they had to redo my account and that has delayed things.  I hope its up in the next day or two. 

Have a great day.

Warm-up

You have 20 people competing with only one kayak.  If each racer takes 3 min 45 secs on average with a break of 5 min between each racer, how long will the whole race take.  This is based on a real race run yesterday in the village.

Warm-up

You are going down river with 20 people and 3 boats.  You need 60 gallons of fuel for each boat and fuel is $6.35 per gallon, how much money do you need to budget for the gas?  The price is what people pay out here.

What is Interactive? Really?

 I've gotten frustrated looking for interactive math activities for my students.  Looking at several definitions of interactive, most agree there is a two way exchange of information such as between two people or a person and a computer.  The other thing is there is some sort of response involved.

To me, this means that one of the people is not just filling in blanks or something similar.  There is a response involved so they are actually active.

Unfortunately, too many "interactive" activities I find are nothing more than having the student read something and answer questions regarding the reading.  This is nothing more than filling out a work sheet, in my opinion.

One step up are the situations where students work together to fill out the sheets where there are questions which require the discussion, otherwise there is no real interaction.  I love those "interactive" notes but the reality is I don't see them as truly interactive because the teacher tells them what to put in, they write it down before gluing it into their binder.  Its really only a one way flow.

To me, an interactive activity might be to divide pairs of students into groups of two.  Give each student a worksheet with different problems but which have the same answer.  This requires them to talk to each other to see if their answers are the same or if they are different, why.  This meets the definition of interactive.

I'll even accept an activity which has students physically doing something such as rolling dice, interviewing people to collect data, or completing an online lesson which requires them to answer questions as they work through the lesson such as many of the lessons at Khan Academy.  Students are not just passively learning.

Many sites such as Kahoot, or IXL actually meet my idea of interactive because the computer provides a response to the student.  In Kahoot, students are told immediately if they were correct and at the end, the computer provides them with the correct answer and shows how many chose each possible answer.

On the other hand IXL does more than just show the correct answer.  They show how to do the problem correctly. In the past, I've told my students stop and check the solution to see where they might make a mistake.  It hasn't worked so this year, I'm going to require they do the work on a separate sheet of paper and then write the corrected version next to it before turning it in.  Too many just want to move on.  I think this will allow students to slow down and learn to examine their work.

Interactive activities require the student to be active, taking part in their learning, otherwise, they are not going to learn as well.  You may not agree with my viewpoint but we can agree to disagree.  My attitude on what constitutes interactive learning has been changing over the past few years.

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

Torus Shapes and Food.

 Doesn't that look wonderful?  A luscious chocolate doughnut whose shape is perfect for a math classroom. 

I was at a conference back in July in Denver and as part of the celebration of science, they arranged for a food truck to be right outside the hotel.  Of course the food truck sold doughnuts because its basic shape if of a torus. 

Imagine bringing a box of doughnuts to class one day with the idea of using them to teach tori in the classroom.  Something students might get into but doughnuts are not the only food item we eat with a torus shape.

People often enjoy a toasted bagel with cream cheese.  Check it out before you slice it and notice it has the correct shape for classification as a torus.  Another thing are onion rings you get with your hamburger.  Those onion rings have the same shape as a torus.  Wow, you could serve quite a few food in this particular shape. If you extend foods out to include those with a toroidal patterns, you can include so much more.  Apples and tomatoes actually meed the definition for a toroidal pattern. 

Once you've introduce this topic along with examples of foods that meet the criterial for toroidal patterns, ask them to think about and decide if the following foods can be classified this way:
1. Calamari
2. Tortalini
3. Canned Pineapple
4. Bialys Pasteries
5. Bundt Cakes
6. Pears
7. Dried apples
8. Jelly Salad molds.
9. Round Pretzels
10. Samosas
11. Cherrios
12. Sliced Olives.

As they think about the above list, they need to keep in mind, the definition of a toroidal pattern.  When they decide yes or no, they need to provide justification for their answer.  This way they show their thinking when classifying them as one or not.  As a reward, students can eat the items when they are done.

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

Imaginary? Really?

At some point we teach students about imaginary numbers.  We define it as the square root of -1.  We show how the pattern repeats every four times it used but are we ever able to provide students with a real situation.  I've been told it is used in electronics but I really have no idea how so I went on the internet to find the answer to this question for the next time I have to teach it.

i is considered an imaginary number,  one used for finding roots of negative numbers, quadratics, etc. 

One reason I didn't realize imaginary numbers appear in electronics is because they use the letter j instead to denote the same thing because i refers to current.  It is primarily found with alternating current which changes from positive to negative sine ways.  Being able to use both real and imaginary numbers help combine alternating currant without being electrocuted.

Imaginary numbers can help describe the movement of shock absorbers in cars as they go over speed bumps, or in modeling how fluids travel around objects in their way, analyzing economic systems and structures using riveted beams, modeling light or water waves, and so many other applications.  Usually, the use of imaginary numbers occurs more frequently in higher level maths.

In addition, you will find imaginary numbers used in signal processing, cellular technology, wireless technology, biology, and radar.  Furthermore, the imaginary number has made an appearance in fictional literature such as The Da Vinci code and The Imaginary by Isaac Asimov where the imaginary number helps describe the behavior of a certain species of squid.

So now when my students ask me about the real life uses of imaginary numbers, I will have an answer.  Let me know what you think, I'd love to hear.  Have a great day.

Math and The Fashion Designer.

 . Inevitably. there are one or two students in every school who dream of designing clothing and starting their own line.  I remember a girl I went to school with who created and sewed her own stuff.  She had talent but most kids now who want to design clothing, know little about the math involved in it.

I came across a presentation or two dealing with this topic when researching the math of pattern making. 

This slide share presentation has some great information and activities.  This packet covers the cost of making the piece so they know how to do it and how to calculate profit, patterns and material, sketching the design using various geometric shapes, parallel and perpendicular lines, symmetry and congruence, types of tessellations, creating prototypes, measuring, fractions, the fashion show itself and refreshments, filling orders from the show by making clothing, and the cost of tailors who make the clothing.  Everything from start to finish and it does include the answers.

This entry is done by a fashion designer explaining how she uses math in her field.  She explains you have to know where measurements are taking you can provide the measurements at the waist, neck, back, etc so someone can make the sample garment, then add or subtract from the sample garment so it fits well.  In addition, designers have to know geometry well so they can create flat pattern pieces which when sewn together make a 3 dimensional garment as well as knowing where to make changes in the pattern piece if needed.

So these two entries show there is math from start to finish.  Think about it.  A designer takes a flat one dimensional drawing, usually one for the front, one for the back and possibly a side view, scales it up to a finished product while determining the shape and size of each piece, creating the pattern, making it and fitting it.

It is not an easy thing to do.  I am not a designer but I can take a historical drawing and create the pattern pieces for it because I know how they used to put together things.  They do things differently today such as in the 1870's a sleeve was usually made out of two pieces to create the correct look but now they use one piece.

I'm off.  I hope you liked this.  Please let me know what you think, I'd love to hear.

Sorry

Due to new computers not being available till today and the local internet provider having problems getting my home internet up and running, I have been without any till today.  I will be back to normal tomorrow.  Thank you for your understanding.

Airline Mles

  I have been doing so much travel this summer that I wondered about air miles and the various plans.  The airline I use as much as possible guarantees a minimum number of miles for any trip, adds bonus miles, gives miles when you use their credit card and the miles never expire.

I know not every airline is like that.  So I was thinking that having students compare various plans from an assortment of airlines both American and International would make a wonderful project.

I know that some airlines have an expiration on mileage while others charge more miles to travel.  I've used some of my miles in an emergency when my sister died, so I could see her before she passed and I needed to get family members to the funeral.  Other times, I've used miles to get someone to a city where the connecting flight began because it was cheaper.

But what restrictions exist with other airlines in regard to using miles?  How many points do you get when you travel?  Do you get more if you pay for a first class ticket rather than being upgraded?  Do you get more points based on the class of the ticket?  I discovered when I traveled on a partner airlines, the class determined how many miles I got.  I ended up with 75% going over but 50% coming back because of the different class. 

When I traveled another partner airline, I got all the miles both ways.  Something to check up on. See how many partner airlines you have access to and what their rules are on miles.  Sometimes, you are not sure until you have traveled.

So this leads to a wonderful project where students investigate the mileage plans of various airlines, check out their partners, find out how many miles are needed to travel to certain places both coach and first class. 

The information can be prepared as a written report as if they were reporting back to their employers on the topic.  The report can include things like graphs comparing the minimum amount of miles needed to travel certain places, miles earned, etc.

Many people are not aware of how their mileage plans work.  Let me know what you think, I'd love to hear.

E-Books and Math

I purchase e-books because I do not have space any more for the hard copy books.  I still order some so I really need to go through some of the books I have now to get rid of so I'll have space for those.

On the first of the month, Amazon always posts the new books for the monthly deal where instead of paying $12.99, you only pay 2.99.

The other thing is that Amazon lists the pages for each book they sell in digital form.  The page number might be at the end of the description or it might be at the bottom in the part that tells you how big the book in electronic size.

Its important to check the size of the book you are buying because if you are like me, you really do not want to spend $5.99 for a 28 page book.  Don't look like that!  I've seen cases like that.  Or $2.99 for a book that's 287 pages long.

I seldom buy a book that has less than 100 pages unless its a very specialized book and I try not to spend over about $6.00 per book because I feel a digital book should cost less than the print copy not the same or more.  A digital book is so much easier to publish, requires nothing in terms of actual printing, and takes less to produce.

There are several math opportunities available a math teacher could implement in the classroom on digital books.

1.  Create a list of books on sale such has those on the monthly deals list at Amazon.  Have students check every book for the print price, original digital price, and the sale digital price.  Students can analyze to determine the percent difference from the print to original digital price is.  They can then determine the percent mark down from the original to sale price of the digital version.

2.  Create a list of digital books on a topic.  Have students look at the number of pages for each ebook.  Have them justify which books they would purchase and why.  One time, I projected the description of two different digital books on toilet training your cat on the board.  We talked about the prices, I think one was $4.99 for a 15 page book while the other was $2.99 for 32 pages.  I asked which one they might buy and why.  I accepted neither as an answer as long as they could justify it.

3. In addition, students can compare the prices of one book in digital form, hardback, paperback, in used and new versions.   The reason behind this, is many used books may be way less but by the time you've added shipping into the cost, the used version is not any cheaper.  Students can create a list of prices for used books broken down by condition before turning the information into various graphs. 

Its important for students to learn to do comparisons based on real world data.  I find the digital world interesting because anyone can publish books now.  In addition, many authors price their digital version higher than the hard copy version.  That I do not understand but it happens and the technical books tend to be more expensive in digital form, just like they are in hard copy.  You might want to let students check it out themselves and then make suggestions on why this happens.

I did take a break from fashion because it is the first of the month.  I'll get back to fashion math in a day or two.  I am getting ready to head back to work so I get distracted.  Have a great day and let me know what  you think.