Flipgrid in the Math Classroom.

In case you haven't heard of Flipgrid, it's a video discussion app/program that allows the teacher to post a topic and students to respond via video.  I've seen it used in a class I took but it was not a math class. 

I'm always interested in finding ways my students can work on using their mathematical vocabulary and improving their ways of explaining or justifying their work.  Flipgrid has a wonderful pdf filled with ideas for grades kindergarten to seniors.

Some ways to use Flipgrid in the math classroom are:
1.  Have students explain how they solved the weekly problem and providing the answer. They could also explain how they solved any problem out of a list of possible problems.
2. Allow students to post problems for other students to solve.
3. Post a find the mistake picture and have students discuss where the mistake was made.
4. Post a would you rather math problem and have them discuss their choice and justification.
5. Create a math help line where students post questions and others answer those questions.
6.  Ask students to compare and contrast mathematical ideas such as adding and subtracting, rational and irrational numbers, linear and exponential models. 
7. Have students develop word problems based on topics that matter to them.
8. Ask students to take current events and turn them into word problems.
9. Have students collaborate on data collecting problems.
10. Ask students to provide current contexts on positive and negative numbers, fractions or any other math topic.  Or they could model solutions to determine the amount of emergency supplies needed after being hit by a hurricane.
11. Have students research a mathematician and share the information via Flipgrid.
12. Have students report back on their thoughts after completing an activity either at Geogebra.com or Desmos.com.

Flipgrid provides an opportunity for students to develop their ability to discuss mathematical topics while doing it in a way they are comfortable.  My students love taking selfies of themselves, record videos, create snapchats and Flipgrid uses this to become part of the classroom.

Let me know what you think. I'd love to hear.  Thank you for reading.  Have a good day.

The Mathematics of Pyramids.

  The cool things about pyramids is that they are not found only in Egypt anymore.  You find them in Las Vegas, and other places around the world.  The even cooler thing about this topic is that you can use Google to bring up actual pictures to give students a better feel for the actual size.

Photographs are nice but they are usually taken from a distance so you get the whole picture but when viewed using google, you get a much better feel for its size.

As you know, a pyramid is basically a square with four triangular sides also known as a square pyramids.  The most famous one is the Pyramid at Giza but others include The Nima Sand Museum in Japan which is a cluster of 6 different pyramids, the tallest of which is over 69 feet tall with a 56 foot base.  The tallest one was built to house the worlds most functional sand timer measuring a year span and is flipped every year at midnight of the last day of the year.

In addition, the Louvre has three pyramids, the tallest of which is 71 feet tall with a 115 foot base.  It opened in 1991 and provides an interesting juxtaposition to the older architectural style of the Louvre itself. Long Beach University has its Walter Pyramid, a monument to athletics as its 18 stories has seating for 4,500 but can hold up to 7,000.  In addition, there are two other pyramids in Russia, one designed for a religious gathering while the other is for entertainment.

We all know the the formula for the volume  of a pyramid is 1/3 x base x height where as the formula for the surface area of a pyramid with regular sides is base area + 1/2 x perimeter x slant length.  In regard to the Pyramid at Giza has some fascinating math associated with it. 

First is the golden ratio (Phi) which is the the only mathematical number whose square is one more than its original number.  Its the number found in nature, its ratio being pleasant to the human eye.  Apply the Pythagorean theorem to these numbers and you get the values which are similar to the golden triangle.

Another idea is that the pyramid could have been based on Pi because the theoretical numbers are within 0.1% different from the Great Pyramid.  Either way the mathematics involved in these claims is fascinating.

Just think with a little work, students could design a Google Tour focusing on these and other pyramid shaped buildings including their volume, surface area and other pieces of information.  They could include a map with pins for each site.  So many possibilities.  In February, I'll be giving a talk on integrating Google into the math classroom, so this would be perfect as an example.

When I get this project done over Christmas break, I'll share it with you .  Have a great day and let me know what you think.

The Mathematics Of A Pyramid Scheme.

  As math teachers we spend so much time explaining the basics of mathematics as determined by the curriculum but what if one day, we took a break and explained the mathematics behind various money making schemes such as the pyramid scheme or a Ponzi scheme. 

I've occasionally had students come in with letters or emails from lawyers or countries claiming they were due to inherit a ton of money or they won some sweepstakes they had not entered.  Those are easy to expose but the other money making schemes are harder but they still make a great topic in math classes.

It turns out the pyramid scheme area covers several types of scams including one that is not as common as it used to be.  The chain letter, the one where you'd get a letter from a friend or relative stating if you send this out to x number of other people, you'd get so much money in a certain amount of time.  Each level brings in twice as many as the previous level, if two people send back money.

Another more common one is the modified 8 ball model in which a person does not get paid until they have recruited enough people to have established several levels.  The idea is that you recruit two people, who recruit two more each who then recruit two more people each so there are three layers with 8 people involved.  At this point, their participation fee is given to the first person. So if the fee is $1000, you just made $8000.

Both schemes use a geometric progression to grow but there is a fine line between legal and illegal in that the people who join have to get merchandise equal to their investment. The actual math formula is well stated here but it boils down to about 88% of the people will loose their money because those past a certain level.

A new scheme is the two up model. The sales earnings from the first two people you recruited goes to the person who recruited you.  It isn't until the third person, you start making a profit.  This involves a geometric progression of three times rather than the two of the previous two.  Unfortunately, those at the very bottom usually do not make money.  In fact about 67% of the people involved in this one, do not make any money.

The formula used to calculate the above averages is the geometric series formula from calculus.

Now for the Ponzi Scheme. This is the people are more likely to be familiar with due to some of those shows that focus on con men who earn millions of dollars before getting caught. The most well known is Bernard L. Madoff who conned people out of over $50 million dollars but it was named after Carlo Ponzi the originator of the scheme.  In 1920, he collected almost $10 million from 10,550 people but only paid out about $8 million. 

The idea behind the Ponzi scheme is that you play upon the greed of people by offering extremely high rates of return which cannot be met so you take the money from later investors and give it to the earlier ones. The mathematics is rather complex but shown here in wonderful detail. 

Lets just say several calculus equations are involved in creating a mathematical model of the ponzi schemes.  The author of the paper, indicates that based on the zeros, it is possible to  determine what is going on with the fund.  If there are no positive zeros, the fund has a positive balance and  is solvent.  If there is one positive zero, it means the fund has collapsed and two positive zeros indicates the fund has become negative but will become positive later on with a bailout.

This explanation is a bit easier to follow but still involves a certain amount of calculus.

Let me know what you think.  I'd love to hear.  I find the mathematics of pyramid and Ponzi schemes fascinating.  Tomorrow, its the mathematics of regular pyramids.

The Brain and Videos

Last week I wrote about a teacher who is telling students that people cannot learn from videos and that teachers in college never use videos.  I suspect her attitude came from the fact she has never learned to be an active watcher.

I found information on how the brain processes the information from a video.  The first element to consider is cognitive load. It has been suggested the memory is made up of several parts.  The sensory memory which is transient and it collects information from the environment.

Information from the sensory memory may end up in the temporary storage or processed in the working memory which has limited space.  The processing is a precursor to encoding the information into long term memory which has unlimited space.  Due to the limitations of the working memory, the viewer must be selective about what they choose to remember.

Cognitive load is composed of three parts.  The first part is the intrinsic load which is determined by the amount of connectivity felt by the viewer.  The second is the germane load is the amount of cognitive activity needed to reach the learning outcome and the third part is extraneous load or the material that is not directly associated with the topic.

In response to this, four practices are recommended to make video learning more effective.  The first is signaling or cuing which uses on screen text or symbols to indicate important information.  Signaling may be by a few key words, a change in color, or a symbol to draw attention to a part of the screen.  Next is segmenting which is the chunking of information so they have control over the flow of new information.  This can be accomplished by breaking the video into small pieces with questions sprinkled throughout so they cannot move forward until they've answered the question.

Another recommendation is weeding or getting rid of any extra material such as music when the person is talking, or extra animation.  It means to eliminate any extra things that can make it harder for the listener to decide if its material they should learn.  The final is matching modality or using both audio/verbal with visual/pictorial to share the new information.  An example of this would be to show a process while explaining it so they have both channels engaged.

So when choosing videos, keep them short of no longer than six minutes.  Make sure the voice over is done in a casual conversational style rather than formal language  because the conversational style makes students feel as if they are partners.  The narrator should speak at a normal speed with enthusiasm. 

For students to get the most out of videos we need to teach them active learning skills such as guiding questions so they know what is important and to help them pay attention rather than switching into a passive watching style.  Make sure the video has interactive controls so the student can rewatch segments as needed and so they have control of the speed at which they move through the video.  Integrate questions into the video.  Research shows that embedded questions increase the understanding and retention of the material in the video.

So videos can be an effective part of the classroom as long as they meet the above criteria so a to meet student learning needs. 

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

Warm-up

Warm-up

Actively Watching Videos

I include annotated videos as part of my instruction because I believe students need to learn to gather information when they watch videos.

Unfortunately, I am fighting the attitude of my students who believe they cannot learn from videos and a teacher who told them point blank that people cannot learn from videos.

She even spoke to me about returning to straight lecturing because professors in college do not use videos, they lecture and our students shouldn't use videos.  I decided the other teacher is wrong because I've taken distance classes which incorporated videos as part of the regular classes. Furthermore, I've gone to YouTube to watch videos to learn to do things like soldier. 

The other day, I spoke with my students about videos.  I began asking students if they ever watch a You Tube video to learn something.  They answered no.  I asked if they ever watched videos for fun.  To be entertained.  Of course they said yes.  This lead to my explaining about mind set.  When they say they are unable to learn using videos, they have a closed mind.  I told them, they can learn but they do not know how to do anything other than passively watch videos because they watch videos with the expectation of being entertained.

The videos I assign are focused on a specific topics with questions sprinkled throughout so they have to stop and answer the questions before moving on.  I took time to go through a video, explaining how I would watch it more actively.  I talked about what I would write down as notes, which material wasn't as important, and how to review the notes I took to find the answers to a question. I told them we'd have another lesson after Thanksgiving to help them learn to watch actively, rather than passively. 

KQED has a column on the topic in which the author discusses how her students entered "TV mode" when watching videos.  In other words, they didn't pay attention to the content, they focused on the accents, or hairstyle or voice.  She had to do things to help them learn to look at the content.

She recommends a teacher pre-watch the video, edit it so it is in smaller chunks, insert questions, comments, or commentaries, and prepare guided notes before showing the video.  The guided notes have blanks so students know what is important to write down.

Before students watch the video, activate prior knowledge and give them a reason to watch it.   During the video, its good to have the video pause often so students can process information and so they can fill out the guided notes. If watching the video as a group, teach students to Watch, Think, Write.  They watch the video, think about it while possibly discussing it, and then writing the information down.

When the video is over, students can create concept maps, use the information to answer a question or solve a problem, find a video clip that clarifies something from the original video and share it with the class.

The BBC recommends the instructor introduce the video to give them a purpose for watching the video.  In addition, it is suggested that videos should not be any longer than 15 minutes due to student attention span.  It is important to include questions so students know what to focus on when watching the video.  These questions turn the passive watching into active watching.  Also let students know they can rewatch a video if they didn't get all the information the first time.

It is good to know that my conclusion of passive versus active learning is correct and that I need to instruct students in active learning when watching videos.  Please let me know what you think.  I'd love to hear. 

Happy Thanksgiving

Print vs Digital Books.

  My classroom still has a full set of print books from several years ago.  My school looked at digital copies but the digital editions would have taken most of the memory of the iPad and would only be good for one year. 

I figured out the cost over 5 years would have been the same.  At the time, digital textbooks were in their infancy.  I've heard they've improved but I was curious as to which is better or if we no longer need the textbook in its current form because of the amount of information and activities available on the internet.

According to a study done by Nielson, most students still prefer their textbooks in print form rather than digital because they don't have to worry about battery charge, its easier to highlight or make notes in the margin, and students have better cognitive mapping with the print.  In addition, most students prefer to take notes using paper and pencil. 

The downside to print is the weight.  Its a strain on the back when a student carries a backpack load of books.  In addition, students who purchase digital versions, do not face the bookstore being out of copies and the digital version is often less expensive.

According to another study, it appears that the print books provide spatial-temperal markers  as people read. The paper and turning the page helps with memory because it makes it easier to remember where you found the information.  Digital devices do not provide the same instinctive markers making it harder to remember where you read something since you scroll to move a page.

In addition, there have been studies done where undergraduate students were given the choice of either paper or digital copies to read 5 different books under different conditions.  At the end, those who used paper copies, scored higher except when disturbed while reading and the two groups scored about the same.  The authors felt that reading a screen is hampered by psychological reasons rather than technological. 

The authors conducted a repeat test two years later to find out if this could be overcome.  They discovered it can be if the person tested begins by preferring the digital format. Most of the research done has been on digital versions of textbooks but these do not address the newer versions of interactive textbooks such as ibooks.  The research I've read, indicates that if a teacher is comfortable with digital books, students are more likely to use them and be comfortable with them.

Personally, the biggest thing I've found with print vs digital books is that my students are not familiar with them and do not know how to read them using "highlighting" or "sticky notes" functions that come with the books.  I have to take time to teach them.  If I choose to use an interactive iBook, I've had down loaded to the iPad, I have to run a class to instruct them in using all the parts so they have a better experience.

I honestly believe students who are experienced with digital books, are used to reading the books for entertainment, so they read passively allowing themselves to be amused.  Yet I believe it takes a different set of skills to read the book actively for information and most high school teachers assume a student already knows how to do that.  I disagree with that.  I think we do need to take time to instruct students in methods of active reading to get the most out of any digital book.

Let me know what you think. I'd love to hear.  Thank you for taking time to read this.

Kahoot

  One of the middle school teachers told me about Kahoot toward the beginning of last week.  He indicated it went well and the kids really loved using it.

Kahoot is a game based site where people create a list of either true or false questions or multiple choice questions on a topic

Students bring up a site which provides entry to the game using a special code that was generated for this event.

The questions are asked, the timer goes off and the idea is to be the first one in with the correct answer.  When all the answers are in, the results are revealed including number of people who selected each answer and which is the correct solution. 

I didn't have the Kahoot app for my iPads but it worked via the Safari browser which was really nice.  Some sites insist you have the app and will not allow you to access the web based version if it recognizes you have an iPad.  It took a bit to get everyone in but once the game started, their level of enjoyment took off and everyone, I mean EVERYONE was engaged. 

As soon as the first game ended, they begged for another game.  When the bell rang, they didn't want to quit. I promised we'd play again on the next Friday class in two weeks.  Now I have motivation for them to get their work done and to learn the mateiral.

I loved there were premade games because I don't always have time to make one.  In addition, I can find a game that is appropriate for the level of my students.  This pre-algebra class is made up of students who are way below their grade level.  I've just started them on learning GCF, LCM, and equal equations.  They struggle with understanding GCF and LCM but once the game started, they tried.  No one got frustrated or gave up.

I found two games from 3rd and 4th grades which worked perfectly for the majority of students.  In general, no one student kept the lead throughout the game.  It switched back and forth because students wanted to be first so they selected the answer, then realized it was wrong and laughed.  Even my special needs students participated.  One is at a second grade level and she had the lead at one point.  At the end of the game, she was proud of her score. 

The week after Thanksgiving, I am going to introduce adding fractions with the same denominator and schedule a Kahoot game at the end of the week so students are motivated to learn and be ready.  Yeah, another tool in my tool chest. 

Let me know what you think.  Have a great day.

Thanksgiving Math

  Elementary teachers have it a bit easier than high school teachers because they can find more Thanksgiving math activities.  On the other hand, too many of those activities are actually worksheets.

Worksheets are fine but they are not very exciting for most of our students who have been raised in a more digital world.

I found a few activities that go a step beyond the usual plug and chug worksheets. I found a few "Thanksgiving logic puzzles" but they were like the emoji math but it used turkeys, cranberries and other traditional foods.

There are a few activities out there that require a bit more.

First is the Gobbler's Dilemma from  Matholicious which looks at opening the store on Thanksgiving day to take advantage of the Black Friday after Thanksgiving sales.  The activity requires students to create a payoff matrix, make conclusions based on the information, and see how the payoff changes over a number of years.  This site is a paid site but they give you a 30 day trial period so you can try out lessons before committing to the whole year.

I've visited Yummy math before.  This year, they have two or three activities which will interest older students while teaching them math.   You can begin with Macy's Thanksgiving Parade, which is televised nationally. This activity starts with a bit of history, then shows the parade route so students have to calculate the length of the parade route in meters, estimating the size of a balloon from a picture and from graphic information, along with some nice real life distance, rate, and time problems.

The second activity involves preparing dinner on the big day. It has students compare fixing the turkey in three different ways, baking, frying, or cooking on a barbecue grill.  For each method, there is information and questions and ends by having a student fill out a chart which includes their opinion of the best way to cook it.  I'm glad to see this chart because our school decided to serve real turkey instead of turkey rolls, so most teachers have volunteered to cook a turkey including myself. 

I am a vegetarian so I'm not very practiced at cooking turkeys.  I bought those oven bags to keep my oven clean and to cook the turkey faster in a disposable pan.   The information in this activity tells me to cook the turkey for 15 minutes per pound.  Yeah.  I am all ready.

Then finish off with a nice activity on Black Friday deals offered the day after Thanksgiving.  The activity specifically looks at the savings a person can get on  a big screen television, X-box limited console and a huge Legos pack if they are the first in the store.  To wrap this activity up, the student looks online for great deals to share with the class after the break.

There is always the age old activity of having students figure out the cost involved in preparing a turkey dinner.  Here it is best to use local prices rather than those found in the city.  I was at the store where I saw turkey advertised for $2.95 per pound.  Cans of pumpkin for a pie runs around $5.00 for a large can.  I hope you liked it.  I"m off to bake a turkey and a pumpkin for the pie I'm contributing to the community celebration.

Have a great day.

Warm-up

Warm-up

Math Museums

  Up until yesterday, I didn't know there was such a thing as math museums.  I knew of art museums, car museums, train museums, historical museums, etc but never a math museum. 

One I found yesterday was the 3DXM Virtual Museum filled with all sorts of mathematical art.  It has material on minimal surfaces, famous surfaces, conformal maps, plane curves, fractals, space curves, polyhedral and math art.

Each topic has a variety of drawings for each type along with the mathematics involved.  The pictures tend to be in color and appear three dimensional.  It isn't until you get to the math art section that things really come together to show math in a different way. 

I checked out the work of several artists and their work is absolutely breathtaking.  The rose shaped parametric surface, the Kleinian double spiral, iso construction, and Tori reflections are spectacular and if shown out of context would be taken as straight artwork.

This site is the type of place students can go and just explore while having fun checking out the interactive elements.

Other places that are not so much museums in the traditional sense of the word but have interesting things to offer both students and teachers.

1. The Exploratorium in San Francisco offers activities that can be done with your students.  One suggestion is creating a math walk locally.  The site offers instructions for creating one locally for people who are not sure how to go about setting one up.

2. The Smithsonian has some nice mathematical opportunities available on line.  There is a virtual exhibit on Slates, Slide rules, and Software a history of mathematical teaching in America.  In addition, there are several videos on the math of prehistoric climate change, or the math involved in the fish population.  There are quite a few stories and articles all associated with math.  So many items to integrate into class.

3.  Finally is the National Museum of Mathematics in New York City does not have much in the way of online materials but it does have a page full of videos made by several different mathematicians.  If you want to see what topics are covered by checking out the map of the museum.

Have fun letting students explore some new places filled with mathematics.  Let me know what you think.  I'd love to hear.

Exploring Math

  The idea for this column came from a tweet by Tina Cardone.   The idea is to create an ongoing project filled with four elements designed to have students do one activity from each column each quarter.  At the end of the quarter, the students report back.

She basically created a choice board with choices to play by doing math art, math comics, or a puzzle,  research either a career, a mathematician, or find an article on math to read and summarize.  There is an explain section where the student explains a new math topic discovered in the last 100 years, or explains about an ancient math system, or an unsolved math problem.  The final section has the student try a math challenge problem, or problems from two other sources.

The play column is in there to show students that math can be fun and is not always answering problem after problem after problem.  So if you want to create your own, where would you go to find comics, art, or puzzles that students might enjoy.  You could borrow her version or you could personalize it to make it more relevant to your students.

Here are places that offer math comics that are actually rather funny.
1. Comic Math filled with samples and links to various mathematically based comics both popular and lesser known ones.
2. This list has comics based on topic.
3. This is a list of 7 places with great math comics.

Now for the art possibilities:
1. Math art for kids has some great ideas including creating a city scape out of the numbers of pi.  There are 21 different suggestions listed here.  Even though its for kids, I think some of my high school students would enjoy them so I think I'll try to find time to incorporate them into my class.

2. Smith Curriculum has some great art projects including the Pythagorean Snail based on the Pythagorean Theorem.  I should try this in my geometry class since some of the students are more artist than anything else and would rather draw than do school work.

3. To see works check out the Virtual Math Museum with some fantastic art based on mathematics.

This site is filled with mathematical puzzles that students might find interesting.

Check this site out as it lists both male and female mathematicians and their ethnicity.  Its quite a list.

My students are not ready for something like this but I like the idea of playing with mathematical art to add another layer to my classroom.  Let me know what you think.  I'd love to hear.

Finding Errors

  As I mentioned yesterday, its hard for students to find the error they made if they do not get the correct answer.  I've been wondering about techniques I can include to help students learn to find their errors.

It seems that once the student has completed a problem, their mind shuts the door on it and moves on because they are finished with it and don't need to check it.

One suggestion I ran across is to have a poster in the classroom for the top 11 errors made in math calculations hung somewhere in the room so they can check it before they move on.

1.  Did not distribute the outside term to both terms inside the parenthesis. This includes not distributing the negative sign with the number.

2. Multiplying by 2 instead of squaring.  In other words they multiply by the exponent, instead of applying the power.

3. Adding instead of subtracting or vice versa.

4. Adding instead of multiplying or vice versa.

5. Misplacing or loosing a decimal.

6. Making a rounding error.

7. Forgetting to carry a number or to borrow.

8. Forgetting to change the inequality sign when dividing or multiplying by a negative.

9. Making a mistake when cross multiplying ratios.

10.  Making a mistake when adding/subtracting/multiplying/dividing a fraction.

11. Omitting units or incorrectly converting units. 

I think I'm going to run this list of common mistakes off and give each student a copy so they can use it to double check their steps.  Of course, I'll have to model its use but if I use it regularly, perhaps they will choose to use their list.

It is also suggested that the teacher change the way they identify mistakes for students.  Rather than saying  "You made a mistake", say "I'm glad you made the mistake, it means you are thinking about the problem and you can learn from it."  I tend to let the student know they missed a step when solving it, so go back and check to see if they can tell where they missed the step. 

In addition it is good for the teacher to make a mistake, correct it, and let the students know what the mistake was and why they did it.  It shows that teachers are not infallible. Teachers are human.  Too often students are under the mistaken impression that math teachers are extremely smart, like Einstein.  Its important to show them we are human.  Make it normal to look at mistakes so they are no longer something to be feared but celebrated.

When a student makes a mistake, it is important to correct it but also to understand why the mistake was made.  By correcting the error and knowing why it was made, it gives the student a personal sense of success. Furthermore, the type of the mistake provides an assessment for the teacher.  The mistakes let the teacher know, what has not been mastered yet.

In a sense, this is something that should be started in elementary but it isn't always so it is necessary to work with students in high school.

Let me know what you think.  I love to hear from my readers.  Have a good day.

Right Answer?

  Too many of my students entering high school are more concerned with getting the right answer.  The ones I've had for a while are moving from right answer to did they do the problem correctly.  Yes, they want the right answer but they are beginning to look at the whole process rather than only the solution.

The same group believes as long as they get the correct answer, its fine, even if the calculations contained an error.  Their answer is usually "So what!  I got it right."  Even if the calculations is correct but they messed up on one thing in the process but still got the right answer, they still say the same thing.  It doesn't matter if it won't work for any other numbers, they don't care because they got the right answer for the problem.

I think its a mind set they get into in elementary school when the problem is right or wrong.  As far as I know, most of the elementary school teachers do not take time to teach students to find their mistakes.  They focus on teaching process and getting the correct answer.

I've heard of teachers moving away from numerical grades into using a rubric based grading system so its not like 67 percent but rather they are not quite proficient in the topic.  It actually sounds more realistic since it eliminates the "How can I bring my grade up?" question.  The last test I gave, many students got upset because I said they could make corrections but the corrections themselves would not raise their test grade.  Making corrections for the test only allows the students an opportunity to retake a similar test.  They don't want that.

I've spoken with the English teacher who said students only want to write one draft, the final draft, before turning it in.  They don't understand that in both math and English, it can take multiple tries to get a finished product that finished and ready to be read.

Because they are so focused on the correct answer, they are unable to take what they just did and do the next problem without asking "What do I do?" or "How do I solve this one."  Its as if they are totally separate from the problem.  I do get students who manage to put it all together to the point they can do all the problems and can help other students learn it.  But too many never reach the point.

Then if you ask them how they got the answer, they give you the "I did the work." answer.  I realize their is a move to explain how they got the answer but when I was in school, we were told as long as we showed the math we used to arrive at the answer, that explained it all.  If I'd been asked to explain how I got the answer, I would have done it by explaining my work at each step.

Right now, I just work on getting the students to look at the whole problem rather than the answer because its easy to make a calculation  error such as 3 x 2 = 5 and get the wrong answer while having completed the process correctly.

Is there an answer?  I don't know because by the time they get to me in high school, they are convinced getting the right answer is the only thing math focuses on.  Let me know what you think.

Comparing costs.

Every math textbook seems to have a lesson or two on unit costs.  Unfortunately, the prices given, even in new textbooks, do not reflect realistic prices where I live in the bush of Alaska.

A Tino's pizza costs around $6 to $8 each.  A pizza you can get at one of the grocery chains runs $21.00 for a $8.00 one on sale in town.  If I use the problems in the book, my students think something is off and don't  relate to them.

Anything frozen that is shipped to the village has the added cost of air freight.  This means I might be able to get a half gallon of ice cream for $5.00 in town but by the time its shipped out here, it costs $11 to $13.00.  Quite a mark-up.

Then there is the difference in prices locally.  For instance, a can of soda runs about $1.25 at the store but various groups buy a 12 pack at the store then resell the soda at $2.00 each.  It is a matter of convenience because they do not have to leave the building, drive to the store, and buy it.  I have no idea what a soda sells for in the lower 48 because I don't usually buy it. 

So in order to give students a real idea of cost comparisons, I have to use local prices.  There are five ways I can have students compare prices.

1.  Comparing different sizes of the same item to see if the cost per unit is consistent, or which one is the better buy.
2.  Rectangular vs circular pizza by weight, are they the same cost?
3. School prices vs city prices.  Our school runs a concession stand that is open during sports activities but not during the day because it sells junk food.
4. Comparing prices between the village and Anchorage so students get a better idea of the markup and cost per unit differences. This is easy to do because two stores in Anchorage do bush orders and have websites listing prices.
5. Compare prices between the two stores in town.  The stores are more like convenience stores but sell guns, bullets, freezers, and just about anything else a K-mart might sell.  I know that sometimes one store has a better price on eggs, milk, or cheese.

Its hard to teach comparing prices using price/unit cost since the store shelves do not have the tags you would normally find.  In a normal store, you might have several different brands or sizes of items but since the stores here are small and limited, they do not carry those tags.  This means my students do not have the opportunity to learn to read shelf tags the way most students do.

They do not get to see the tags showing the price/unit cost in different units.  I've seen two of the same type of product listed with price/product vs price/ounce which makes it much harder compare.  So I have to get creative for this type of comparison in the classroom.

What I do have available is the SpanAlaska Catalogue which allows people to order items in bulk.  I can have students look up various items and have them calculate the price/unit cost using the same unit.  I could have a friend take pictures, send them so I can show students how the tags normally appear.  I have to create the experience for my students so when they go somewhere with the shelf tags, they are educated and capable of using them.

Let me know what you think.  I'm interested.  Have a great day.

Warm-up

Warm-up

Math Girls.

As I checked out graphic novels, I came across a series of novels called "Math Girls". This series is written by Hiroshi Yuki and the first book was originally published in Japan in 2007. 

Since then, the series has sold over 100,000 copies in Japan alone.  Beginning in 2011, these novels have been translated into English and now can be bought here in the United States.

The three main characters are the Narrator who is the protagonist and everything is told from his viewpoint, Miruka who is a second year high school student in the same home room as the Narrator, and Tetra a first year high school student who went to the same middle school as the Narrator.In addition, Miruka lives breathes and talks math while Tetra has a serious case of math anxiety.

As far as I can tell, there have been seven volumes released on the Math Girls.  The first volume titled Math Girls appears to be the first general introductory volume where we become acquainted.  The next three cover equations and graphs, integers, and trig.  The next two cover Fermat's Last Theorem and Godel's Incompleteness Theorems. The final one is a volume 2 of the Math Girls Manga.

The books for the most part are made up of conversations between the narrator, Miruka and Tetra along with a supporting cast.  You see the dialog, with the mathematical information and proofs sprinkled throughout the story. 

Only two books are done in the actual manga style but only one is available in this country.  The rest are only available in paperback rather than ebook format.  I wouldn't mind having a set in my classroom just so students can see that math can be appealing in a written form that is not a picture book.

Check the books out at Amazon if you'd like to see them in more detail.  Let me know what you think.  I love hearing from people.

What is Number Fluency?

  I am always hearing that students need to develop number fluency but what is meant by that?  Number fluency is another way of saying students need to develop number sense. 

Students need to develop number fluency in conjunction with conceptual understanding and computational fluency.  In other words, they need to understand the concept and be about to perform calculations fluidly.

According to one definition, number fluency means a student can compose and decompose numbers in a variety of ways, is able to see patterns in numbers, knows their basic mathematical facts fluently, is able to work fluently and quickly with numbers to solve problems,  and is able to work well with place value and with numbers.

People often wonder why number fluency is important.  Number fluency is the bridge between recognizing numbers and understanding how to solve problems.  A student with number fluency is able to look at the problem rather than focus on basic facts.  They are able to make connections with prior knowledge more easily than someone who has to focus on their calculations.

Too many of my students arrive in high school still skip counting on their fingers rather than knowing their multiplication by heart.  I realize there are calculators out there but if a student has not developed a sense of what the answer should be, they do not know if their answer is even close.  Too many of my students accept the answer from a calculator as the correct value rather than taking time to ask themselves if it is a reasonable answer.

Since almost every student has a mobile device with a calculator, they want to use it rather than trying to remember their multiplication and division facts.  If I allow the use of a calculator, I insist that two students work together and each one run the numbers.  This way if they disagree, they can try to figure out who put the numbers in incorrectly.  This way, I hope it helps them build a sense of the way numbers work.

When it comes to fractions, the first thing they want to do is change the fractions into decimals because they find decimals so much more comfortable to work with.  Unfortunately, there are certain fractions that when changed into decimals become repeating numbers and rounding the value does not help.

Once thing I love to do is to have students analyze problems during warm-up to determine if they are correct or incorrect.  If they are incorrect, students have to determine what the error is and when it occurred during the calculation.  This makes students slow down and think so at least they are developing more of a number sense than what they arrive with in high school.

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

Adding a Dimension To Fractions.

I am teaching a pre-algebra class this year.  I've discovered most of them struggle when adding or subtracting integers. The see the - sign as subtraction rather than a negative number.

I always spend the first semester building their skills before introducing the algebraic element.  This year, I am going to do something a bit different.

Instead of teaching fractions using only positive quantities, I want the students to learn fractions are not always positive.

If I find a piece of material, a remnant, that is 1/4 inch short of the length I need, that would indicate a negative value.  On the other hand, if the material is 1/4th of a foot over, that would be a positive value.

My students entered high school with certain ideas such as you cannot subtract a larger value from a smaller value so you have a negative result.  Like if you write a check for more than you have in your checking account.  They also see -4 -6 and do not recognize it as -4 + -6.  Even after spending two months on it, they still struggle.  I've used chips, number lines, everything I can think of and they still struggle.

I already know they are going to struggle when I write 5 1/4 +(- 1 1/3) instead of 5 1/4 - 1 1/3.  I suspect even having them draw pictures and  using number lines when they begin working with simple fractions, they will still struggle.

When we start the topic, I plan to have them go onto the internet to find ways in which fractions are used in real life.  They'll have to use their own words to describe each situation and provide a picture to illustrate the use.  Too often, they do not connect what they learn about fractions in school with their use in real life.   

Once this activity is out of the way, I plan to use some activities from Texas Instruments with a bit of modification for my students.  The activities range from the general question of "What is a fraction?" to discovering that fractions are equivalent if they are found at the same place on a number line, to mixed numbers.  There are 15 different activities in this unit.

When it comes time to discuss common denominators, I've found graph paper is wonderful for creating models designed to show students why any fraction must have the same denominator to combine.  Years ago, one of my students admitted they didn't know the boxes had to be subdivided into equal parts.

I also have a couple of games on my ipads for students to play so they can practice using fractions in a more fun way. Towards the end of the unit, I plan to break the students up into groups to create a game using fractions.  Once the games are completed, I'll have other groups test the games based on a rubric. 

I hope they have an easier time learning this topic than they did learning integers.  Let me know what you think.  I'd love to hear.

Math and Exercise

  There is now research indicating that increasing physical activity throughout the day can increase both reading and math scores.  It makes sense to me because when I have to sit through a professional development day, I get tired and sluggish.

It appears that regular aerobic exercise in children helps improve their ability to do math.  It has been theorized that the aerobic exercise results in a thinner layer of grey matter which is believed to improve cognitive control and working memory.  Both of which are important skills in doing math.

It is now being suggested that some sort of physical activity occur in the classroom during the lesson or just before to improve brain function.  Even just having them get up and walk around just before presenting an important activity helps increase understanding.

A study from the University of Copenhagen indicates that if the physical activity is integrated into the part of the lesson focused on learning the material, student understanding increases and student scores go up.  The physical activity included a whole body element such as drawing shapes with the whole body or using groups of students to do addition or subtraction problems.

It is much harder to create physical activities in the secondary math classroom but at least one high school has managed it by having students take gym first period before math.  The gym taught includes things like square dancing which requires the complicated movements designed to stimulate thinking.

In addition, if a math teacher notices a student starting to zone out in class, the teacher implements a short burst of physical activity to get the brain working again. This school has seen its reading scores and math scores increase significantly.

Furthermore, regular physical activity is great for improving behavior, concentration, and physical shape.  It can also decrease a person's anxiety prior to a major exam.  Research has discovered there is an immediate improvement in concentration as soon as they begin regular exercise.  Over time, the ability of the brain to process mathematics improves.

I found this to be quite interesting.  I think I might need to include a bit of physical activity in the middle of my classes to increase brain power.  Students take 7 classes per day with only a four minute break between and one 30 min lunch period.  Not really enough time for the brain to get the stimulation it needs.

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

Real life inequalities

  Its hard for people to identify situations involving inequalities in real life because they are not neatly laid out in the same manner as found in textbooks. 

Inequalities are all around us but most teachers spend time teaching students to create problems from written word problems or learning to solve problems without spending much time looking at how these problems appear in real life.

So I'm taking time to look at some everyday situations which are inequalities so perhaps we can expose our students to more problems as they look in real life, not in the text book.

1.  Trucking companies are very aware of inequalities because they have to deal with weight limits set by bridges.  Most bridges have an upper limit for the maximum weight of a truck, its cab, and its cargo so companies must be aware of how much cargo they can load the truck with since the weight of the cab and trailer is static. 

An example might be the truck has to cross several bridges with a maximum weight of 65,000 lbs.  The cab is 21,000 pounds while the trailer's empty weight is 19,000 pounds.  This means the cab and trailer weigh 40,000 pounds so the truck cannot have more than 25,000 pounds of cargo.  The inequality is less than or equal to 25,000 pounds.

Airplanes and trains also have to watch their weight due to fuel consumption based on number of passengers, temperature a few other things.

2. Another inequality problem is when we want to buy something like sneakers and there are several models we like but they each cost a different price.  When we need to know how many hours we need to work to afford them, we look at the cost of the cheapest pair to find a minimum. 

An example might be that we've found three pair of basketball shoes we love priced $140, $143, $148.  We've already saved $40 towards a pair so we'll need to determine how many hours we need to work to afford the cheapest pair.  This is a greater than or equal to type problem because we need to know the minimum amount we need to earn.

3. I worked a job one time where I got paid so much per hour but if I sold at least $250 worth of product each week, I would start making more per week.  I received a set amount of money for each item I sold plus the base wage. I could easily determine how many items I needed to sell to reach a certain salary. 

4. When you take the elevator, there is a maximum weight associated with the number of people the elevator can safely handle.  This type of problem is a less than or equal to because the weight cannot be more than a certain number. In addition, they use an average weight for people to determine the maximum number the elevator can handle.  This means you can have more if they all weigh less then the average or fewer if they weigh more.   The next time you ride an elevator check it out.

5.  Sometimes you find inequalities when dealing with the number of letters placed on a t-shirt or on a piece of jewelry.  For instance, you are charged $0.10 per letter and you have a budget of $5.00, how many letters can you afford and what are some of the things you can have engraved on the jewelry?

These are just a few situations in real life which use inequalities.  I like them because I relate to the situations and so do most everyone else.  Let me know what you think.  Have a great day.

Warmup

Warm-up

Proportional Reasoning.

Proportional reasoning is the ability to compare two things using multiplicative thinking.  Something like when we have students use proportional reasoning to identify similar triangles, or solving for the unknown.  Sometimes we find problems in the book for enlarging photos or scale models.

I've seldom seen it applied to a real life situation until I came across an article applying it to social justice.  There are several types of problems which fall under the general topic of social justice.

1.  Calculating population growth and the associated crime rates. One way to handle this is to look at the population growth for local populations so students can determine when they need to find differences and when they need ratios.  They also need to determine when starting population differences make a difference and when they do not such as in building a fixed number of houses or creating a new electrical grid.

Population growth is important.  I have some cousins who live outside of Washington, D.C. in Virginia.  The area underwent tremendous growth because people could still find a great deal in the cost of housing but traffic became extremely bad when the area could not keep up with the number of cars on the road.

When you start to compare crime rates in the same areas and look only at violent crimes, students can look at ratios and percentages to determine which location is safer based on that criteria.

2.  Look at the representation in Congress versus the population of racial composition by state or over all.  If you wanted, you could take a look at the representation in the state government versus the racial composition of that state.

3. Look at historical immigration rates from 1820 to 2000 to see how rates have changed over time.  You could even look at recent rates to see where the majority of immigrants come from. 

4. Look at the pay people who work in sweat shops make versus the cost the item sells for in stores in the United States.  Calculate the daily, monthly, and yearly rates earned by these workers.

This google site has a wonderful list of places to find additional lessons connecting proportional reasoning with social justice.  In addition, many of the lessons have students applying mathematics to social topics. 

In addition, the October 2015 issue of Mathematics teaching in the Middle School magazine has great information for turning general textbook problems into more personalized problems students can relate to better.

Let me know what you think. I look forward to hearing from people.

Designing Sports Bags

If you every look at crowd sourcing site on line, you'll find an assortment of groups who market the "perfect bag".  Yes, I've invested in a few of which one has gone into production and been delivered while the other one will soon arrive.

I invested in a backpack that allowed me to spend a couple weeks in Europe with only a small roller bag as my check through.  It was great.  I spotted a task which would be perfect in the Geometry classroom based on shapes and nets.

Why no assign students the task of designing a sports bag or gym bag to certain specifications just like designers do in the real world?  The teacher is the manager who assigns the students to design and make a patterns for the sewing room to make the object.

Criteria could include:
1. The finished bag is 55 cm long
2.  The circular ends must have a diameter of 22 cm.
3. The body is made from a single piece of fabric.
4. The ends must be made from the same fabric.
5.  Make sure there is an extra 1.5 cm added to all sides for the seam.
6. The fabric is 1 meter wide.

The student needs to create a pattern with the exact shapes needed.  In addition, the student needs to determine the smallest amount of fabric needed to make the bag.  This means they need to create a cutting guide, just like the ones found in patterns.

A perfect place to insert this type of activity in the geometry class is when studying the nets needed to make three dimensional shapes such as cylinders, pyramids, etc.  I love finding any application I can for real life applications.  In the past, I've had them just create the nets for various shapes but this activity takes it a step further.

There are a couple sites which have wonderful lesson plans for this activity.  One is at the Mathematics Assessment Project, which has everything needed to run this in your classroom.  If you prefer, you could have students design a backpack (rectangular prism) or other type of bag.

Let me know what you think.  Have a great day.

The Moon and Math.

  We are in the time of the year in Alaska, when we will often see the moon for 24 hours or longer at a time.  There are times when it does not set but circles around the sky.  It is an awesome sight.

The moon and its phases offers some great math opportunities for the classroom in geometry and in measuring time.  Two different aspects of something we see all our lives.

Lets start with the relationship between the time of day and the visibility of the moon based on its phase. It is important to remember the moonlight we see is actually sunlight bouncing off the moon.  Consequently, the phase of the moon is determined by the relationship between the moon, the earth and the sun. 

So the geometry involved is as follows.  When the moon is between the earth and the sun, we observe it as a new moon because we are looking at the moon without illumination. Two weeks later when the moon is on the other side of the earth from the sun, the moon is fully illuminated and we see it as the full moon.  The two quarter moons occur in between when the moon is half lit. When all this is happening, the moon is rising as the sun is setting during the full moon.

Add in an activity that uses a spreadsheet to determine the next full moon.  The moon's orbit is 29.5 days long so its a matter of adding that amount to the current time of moon rise.  However, if you want to make it a bit more interesting follow the link to get the in depth equations to create a wonderful sine wave visualization of the moons cycle.  The equation even goes so far as to include the percent of the phases of the moon.  The author has them adjust the equation to make it a bit more accurate.  See if you could coordinate this activity with the science class when they are studying phases of the moon.

Of course NASA has a 128 page book filled with various science and mathematical activities for grades 3 to 12 on lunar math.  It includes a list of math topics covered in this book.  I took a look at some of the activities for younger students and with a bit of adjustment, they could be used in the high school. 

This lesson geared for middle school seems to tie the geometry of moon phases and spread sheet into one nice long lesson.  It has everything needed and all the links for supplemental materials. 

Finally, there is a four lesson unit in the NCTM middle school magazine which integrates science, math, and literacy on using moon phases to measure time.  Lesson one is focused on lunar phases and how they work.  The second lesson focuses on what make a good standard unit because we measure things in standard units such as feet or meters.  The third lesson focuses on the Hopi because their astronomers did a good job of marking time. This lesson includes the lunar cycles based on the Hopi names for the moons.  The final lesson looks at the ways different cultures describe time.

Have a good day and let me know what you think.  I love getting feedback.