Facts and Figures about Times Square's New Year's Eve Celebration.

 

The ball they drop from the top of a 77 foot pole situated in the Times Square area of New York City has some fascinating facts associated with it that provide some great math questions.  There is the first possibility.  The ball travels 77 feet in just 60 seconds so we have a question on its rate of travel.

Another interesting fact is the original ball in 1907 had a diameter of just 5 feet but by the time of the last ball, it had grown to having a diameter of 12 feet.  That is enough information to calculate the volume of both and determine the percent increase in volume since the first one.  

Some other interesting facts associated with the ball is that the original ball in 1907 weighed 700 pounds but in the 1920's it was replaced by a 400 pound iron ball.  Then in 1955, it was replaced by a 150 pound aluminum ball which was replaced about 10 years ago by a ball weighing 11,875 pounds.  This information opens up to calculating change in precent between the balls.  There was a trend of getting lighter but then with the last one, it became so much heavier.

One reason the current ball is so heavy is due to the 2,688 Waterford crystals imported from Ireland and to the 32,385 LED lights covering the ball.  Since this has a 12 foot diameter, it is possible to calculate the surface area of the sphere and then use that to calculate the density of the crystals and the lights.  In addition, the 32,385 lights can create 16 million color combinations.  That is interesting that so few lights can produce so many combinations.

Once the ball has fallen, the city releases 3,000 pounds of of confetti.  The 3000 pounds is made up of around 30,000,000 pieces of colored paper.  Add into that the fact over a million people attend the event in person and everyone produces something like 48 tons of garbage that takes around 7 hours to clean up using a total of 180 people.  These facts provide students a chance to calculate how much trash each person produced.

If you don't feel like using any of this information to create your own activities, there are some out there ready to go.  Yummy math has a couple that are quite interesting.  One activity that deals specifically with ball itself.  The ball is actually a geodesic sphere that is fixed to a frame.  The activity goes into more detail about the arrangement Waterford crystals and LED lights on the dome.  

The other activity looks at the evolution of the ball itself from the original 5 foot diameter wood/iron one to the current one that is more of a wonder/notice activity.  It wouldn't be hard to include a second page with questions that have students calculate percent increases or decreases from ball to ball.  If you feel like looking at the amount of work as defined by physics, check out this page which shows the math involved in finding the amount of total work done by the ball. 

Now if you like, you can have students take the information from this article published in 2019 and have them create an infographic with all the information or assign specific areas for students to investigate in more detail before creating the infographic using that information.  

Due to the virus, this year's celebration will only happen virtually so you can still see it happen from the comfort of your own home.  Let me know what you think, I'd love to hear.  Have a great day.

The Cost of Being "Santa Claus".

Do you remember as a child, going to the local mall where you saw Santa Claus?  He was there everyday between Thanksgiving and Christmas and you ended up standing in line just so you could tell him exactly what you wanted for Christmas this year?

This is one of those seasonal jobs you might want to talk about in class next year.  This job has certain requirements and costs.  For instance, to be successful, a person needs to have a certain look.  

The look includes a proper red suit which can run between $800 and $1200 for a high quality one.  It is possible to pay less but it won't look as good and Santa wants to look their best.  It is important to have at least two suits because people, including Santa, spill things on themselves and little ones can have sticky fingers.  Then they have to purchase appropriate boots, a thick belt with a proper belt buckle which runs between $250 and $400.  Don't forget that one has to properly clean the suit on a regular basis so that costs between $50 and $150 each time.

To stand out as Santa, it is important to customize the suit by adding buttons that run between $6 and $75 each, or keys, or pouches.  Furthermore, Santa has to have the proper colored hair and beard to go with the outfit.  A realistic beard and wig combination will run a person between $1500 and $2000, depending on the type.  By type, I mean is it a short beard, medium, or long one and you would want a custom one so it looks real.  In addition, you'll need to consider spending another $200 or so per year to have the wig/beard sets cleaned and maintained for the next year.

If someone uses their own beard and hair, they have to make sure it is of the same silvery white one sees in pictures and that can run between $100 to $300 per session.  If a person wants to be Santa, it is also recommended they carry Santa liability insurance policy just in case.  The suggested amount is in the 2 to 4 million dollar range for proper protection.  

Then if you want to stand out, you will lay out between $250 and $550 to attend "Santa" School to learn the finer points of being Santa.  These schools provide the necessary skills to make it as a professional Santa who will entertain everyone and learn more about doing a proper job.  Enrolling in one of these schools will improve a person's chances of making more money doing the job.  Since much of the world is digital, there is even one school that provides online classes for potential Santas.  

This is the cost of becoming Santa and yearly outlays to continue being Santa.  So what can a person earn as Santa each year?  Well, that depends on several factors but according to two different sources most professional Santas earn between $5000 and $8000 per year with some earning between $15,000 to $20,000 annually.  If you work at a mall or are hired by a professional photo company to pose with children, the median pay rate is said to be $41 per hour.  The better the Santa, the more money a person can earn.  One article said if a person looks and behaves like a "real" Santa, has had proper training, and has the right attitude, they can earn up to $500 per hour.  Unfortunately, the season is generally limited to November and December so it is considered a seasonal job.  

Maybe next December, you can share this with students as one of those jobs you don't think about.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If most people manage to adhere to their New Year's Resolutions to January 12th?  What percent of the year were they successful?

Warm-up

Last year after Christmas season sales ran around $1.14 trillion.  This year, it is predicted to run around $1.52 trillion.  Find the percent increase.

 

Merry Christmas

The True Cost Of the 12 Days Of Christmas

 

Every year, one of the songs you hear during the Christmas season is the 12 days of Christmas with it's golden rings, drummers drumming, and all the other pieces.  Every year, someone figures out the cost of what these gifts might set someone back.  Every year since 1984, PNC Financial group has published the current cost of obtaining these gifts and compares it with the previous year.

Due to the pandemic, PNC was unable to calculate the cost of nine ladies dancing, ten lords a leaping, eleven pipers piping, and twelve drummers drumming.  PNC uses prices from performances to calculate these items but with performances being cancelled, they are unable to get the current cost.k. Consequently, the cost for this dropped by about 38% since last year.

The list breaks down the cost of each item and explains whether it went up, stayed the same, or went down and why.  For instance, it tells us that it will cost $58.00 to hire the maids-a-milking since the federal minimum wage has not changed since last year but the cost for two turtle doves is running $450 this year, an increase of about 50% since last year.  

In addition, it lists the cost of the 12 days since 1984 when they first started doing it.  This means students can take the data with 1984 being the base year or 0 and they can create a line of best fit via linear or quadratic regression and come up with the equation.  A practical application of data in a fun manner.  Furthermore, at the bottom of the page, they have a graph showing the total cost of the gifts over the years but they also allow you to look at the changing cost of each individual item.

This is wonderful information because students have to read and interpret data.  The information also allows students to calculate the percent change from year to year for individual items, find the best fit equation using regression and a few allow students to use piecewise functions to mathematically explain the line.

Another activity this particular graph allows is for students to take a specific year and use the data to break down what percent of the whole cost each item represents both now and historically.  The information could be used to create bar graphs for each year to show how the individual items have changed or remained constant.  IF you look at the historical cost of the eight maids a milking, it is based on the Federal minimum wage and thus remains fairly constant.  The graph itself looks like steps, so students can see there are situations where step graphs appear in real life. 

This one page offers students a chance to practice mathematical modeling, graph interpretation, data use and interpretation that are all real life.  Let me know what you think, I'd love to hear.  Have a great day.

The Post Office, Christmas, Wrapping Gifts, and Art..

 

It is the time of the year again, when we have to meet deadlines so Christmas gifts will make it to all the relatives in time. The village I live in, finally got the a bunch of mail after almost 10 days of no mail deliveries from Anchorage.  The post office was actually open Saturday due to the huge amount of mail arriving and more came in on Sunday.

According to a release, in 2019, the United States Post office delivered over 28 million packages each day between December 16 to the 21st.  In addition, the number drops to about 20 million packages per day to the end of the year.  Furthermore, it is predicted the post office will deliver over 800 million packages between Thanksgiving and New Years day.  

This year, everything is so much different due to the Coronavirus.  Since so many people are under lockdown and cannot travel, the number of orders made to online shops has increased significantly so the number of packages being shipped is threatening to overwhelm the post office and other shippers. It is estimated that e-commerce sales will reach close to $189 billion for November and December.  This is a 33 percent increase over last year.  The above information can be used by students to figure out how much the e-commerce sales were last year. 

Now for looking at holiday activities one can do in the classroom to celebrate the season.  This site has a wonderful breakdown of the number of toys that must be delivered to children over the 24 hour Christmas Day.  It was done by an Engineer back in 1990 but it would be easy to have students update some of the information to see how things have changed since it was originally written. Once students have updated the information, they can calculate the percent increase or decrease over the years.  The article offers the opportunity to carry out an analysis of the original data.

Christmas time is perfect for an exploration of surface area due to having to wrap presents.  Yummy Math has a great activity that has students comparing wrapping presents using the traditional method where the paper is set to go with the present or wrapping the gift on the diagonal to see which one uses less paper.  The lesson uses both surface area and the Pythagorean theorem during the exploration.

Yummy Math has another activity involves students determining if they have enough paper to wrap a certain present.  The specific wrapping paper chosen is the tissue paper which based on personal experience requires at least two to three sheets to properly cover a present and usually comes in smaller sized sheets than wrapping paper on a roll.  

Yummy Math also has a variation of the wrapping activity where students are shown a picture of a man who is purchasing two large gifts.  Students are asked to estimate how much paper will be needed to wrap these gifts and if one roll of paper is enough. Students are given the length and width of the roll of paper and they have to calculate area, estimate the measurements of the gifts, and are required to make predictions. 

This is also the perfect time of year to have students practice coordinate graphing since there are a lot of free coordinate drawing activities.  At this site, you can get one that allows students to create a picture of Santa Claus.  Math-aids has at least 6 different graphs to celebrate the season with trees or a gift or other ones.  Prefer something cute, check out the well wrapped Penguin for a winter celebration. On Wednesday, I'll be checking out the cost for someone to share the 12 days of Christmas for a love.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If there are 328.2 million people living in the United States and each person eats on average 27 pounds of bananas every year, how many tons of bananas do Americans consume in one year?

Warm-up

There are an average of 18 tiers of bananas and each tier has 16 bananas.  How many bananas are there on this bunch?  Now if each banana weighs 4.2 ounces, how many pounds is in the bunch?

Which Weighs More? A Pound Of Feathers Or A Pound Of Gold?

 

For the weekend warm-up, I looked up the number of feathers in a pound and came across a very interesting answer to the question -"Which weighs more, feathers or gold?"  The normal person will usually answer it with either "Don't they weigh the same?"  or "The gold of course because it obviously weighs more. 

Most people don't realize that the two items are weighed using two different systems.  Feathers are weighed using the "avoirdupois" system which uses 16 ounces per pound where as gold is weighed in the troy system which only has 12 ounces per pound.

Furthermore, the avoirdupois system is used to weigh things like food, people, mail, and equipment while gold, silver, and other precious metals are weighed with the troy system.  So if both the feathers and gold are weighed in the troy system, something strange happens.  The feathers weigh more than the gold.  This happens because one avoirdupois ounce is equal to about 28.1 grams while one troy ounce is equal to 31.1 grams. So a 16 ounce pound of feathers weighs around 453 grams while the 12 ounce pound of gold weighs around 373 grams.  

The whole troy system got it's start in the French town of Troyes and was based on pennyweights and grains of wheat or barley.  Yes, grains of wheat but more on that later.  First a troy ounce has an assigned equivalence of 20 pennyweights and a pennyweight was defined as the weight of a silver penny in England.  Now back to grains.  The grain of wheat or barley is the basis of both the weights in the troy and the avoirdupois systems.  

If you look at the grains as the basis of both troy and avoirdupois systems, you'll find one carat is equal to 4 grains, a troy ounce is defined as 24 grains and a gram is 15.432 grains. Now the thing about the grain measurement is that the troy systems is said to have been based on barley grins while the wheat grain was used as the standard of weight.  Apparently three barley grains equaled four wheat grains.

The question of "Which weighs more? A pound of feathers or a pound of gold opened up the possibility of having students learn more about avoirdupois and troy systems, their differences, and what they are used to weight.  In addition, it opens the door to include some history into the math class and it also can lead the the question of "Is it fair to compare these two items in the way it was done?".  Who knows you might get some interesting opinions from your students.

I hope you found this as interesting as I did.  Let me know what you think, I'd love to hear. Have a great day and enjoy your weekend.

Ways To Have Students Participate Digitally.

Many of us are either already teaching virtually or we will soon be doing that.  I also know that many school districts are switching back and forth between in person and virtually so it is important to know of ways to keep student participation up in a virtual climate.  

It is important to do this because you have students who will always try to be the first one in with the answers and others who will never say a word but for students to learn, they have to participate.

Digital platforms also make it harder to carry out conversations because you can't "see" others who raise their hands, or who just unmuted to talk but the host can see it.  In addition, not everyone has the ability to participate digitally so when we discuss participation, we have to talk about having everyone involved, not just those with access to Zoom or Google. Since most people have cell phones, it is possible for students who do not have good internet or a computer to call in via phone so they can still participate in synchronous learning.

Lets start with ways to get students involved virtually or via synchronous learning.  One way is to provide the material to students ahead of time to read and answer associated questions.  When the online meeting starts, students share their responses with each other while the teacher draws a picture or web showing the flow of the conversation.  At the end of the conversation, the teacher shows students the web so they can analyze it and reflect upon who spoke, who listened, and who used ideas as a foundation to build upon.

Don't be afraid of using the chat feature in Google Meet or Zoom.  One can ask a question, let the students put their answers in the chat box but not send them until the teacher says share.  Teachers can also to check for understanding by having students provide a thumbs up or thumbs down. Chat boxes are also great for having students ask questions, provide answers, or even answer True or False questions and providing justification.

It is possible to do Think-Pair-Share digitally by providing prompts and having students divided into smaller rooms to discuss things and come up with answers they recorded in google documents on on google slides for accountability.  When they returned to the group, volunteers from each group shared their conclusions.  One can take this a slight different direction by assigning small groups of students a math problem to complete.  Once they've gotten an answer, they share it with others using the information in the document or slide.

One can also combine asynchronous with synchronous learning by having students watch videos and completing online activities at home before coming to class digitally. Once in the virtual classroom, students can ask questions, split them into smaller groups for small group work, and clarify misunderstandings.  

As for asynchronous learning, use the question feature in Google classroom to ask students questions and have them answer.  In math, the question might be something like "In this problem would you distribute or divide first?  Explain your answer."  or it could be asking students to ask questions about what they didn't understand in the lesson.

One can also have students create a short video showing how to solve an assigned problem.  The teacher would up load the videos to say google classroom so students can watch each one and they are required to post a question to two other videos.  

It is good to know there are ways to increase student participation.  Let me know what you think, I'd love to hear.  have a great day.

Figuring Out What To Teach During The Pandemic.

 

I  really dislike trying teaching during these trying times.  The district I work for, provided a list of topics I should concentrate on, shorten, or eliminate.  Unfortunately, many of the topics they said to drop are needed to provide a foundation for later concepts.  Although our school is in person, I have many students who are regularly gone due to someone in their family being on quarantine or in jail, or other reasons so I have to consider this when I write a lesson.  In addition, when we are red, we use packets since not all students have computers, or reliable internet.  The nice thing about graphing is there are apps I can have students download onto their phones so they can graph without the internet.

Unfortunately, when I have to send packets out, I have to include detailed explanations to provide additional clarity for textbook explanations.  In addition, I create videos that are posted in google classroom for students who have decent internet and are available on thumb drives for students with a home computer.  Rather than having a complete online class, I have office hours via google, zoom, or phone so students can either attend digitally or call in via phone to ask questions.  If they have e-mail, I will send answers via e-mail if they phoned or sent a question in.

I've taken two classes on teaching virtually, one was on math but both seemed to be geared more towards teaching via Zoom with the assumption, all students have access to great internet. Neither one really addressed the issue of creating packets that can be used by students to learn.  Packets require students to be self motivated and willing to work.  I'm not sure how to have students complete work if they normally struggle in the classroom.  

I have students who struggle in person because they have gaps in their foundational knowledge.  They need to ask questions. They need to make sure they are doing the work correctly.  They need the reassurance they get in person. I don't know how to convey that via a packet.  This is one huge reason I ask myself if they really need that topic now. 

For Algebra I, I can cover the material when they see it in the Algebra II class while those in Algebra 2 should see it again when they take pre-calculus or go to college.  I've come to the conclusion that if I can't cover it this year, they will have another opportunity to learn it. I do wonder if I'm being fair but I do think I'm being realistic about this.

I admit that  for some topics, I'm stressing using graphing to answer rather than mathematical methods because it helps students learn to interpret graphs while understanding the connection between the written equation and it's visualization.  I hear again and again that students need a visual to connect to the equations.  Fortunately, the graph works for many things.

I also realize that no matter what we do this year, no matter what we cover, our students will be have gaps in their knowledge. Perhaps, more than normal.  We try but we can only do what we can.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

You need enough feathers to cover eight more costumes. The first costume took 587 feathers while the second took 625. If one pound of feathers has about 283 feathers, how many pounds will you need to finish the costumes?

Warm-up

 

You are in charge of providing masks for your school.  Everyday, one 50 pack of masks is used by those who forgot theirs or do not have any.  How many packs will you need for the next three months?

More History With Math.

This past Saturday, I checked for new and interesting math apps for my iPad and I came across one put out by the Midway museum in San Diego California.  I was surprised to find something on the Midway but the authors of the app decided to combine some history with math to make it a bit more interesting. 

The Midway Museum STEAM app has combined information on the aircraft carrier with mathematical problems.  Admittedly there is only one math question per section, this is one of the first apps I've seen to combine history with math.

The app covers 6 areas of the ship.  It looks at the enlisted berth where men sleep in triple decker beds, the foc'sle where the anchor is kept, the galley, the amount of food needed to feed the crew while they are underway, helicopters and the flight deck. 

Each section either has a short video to watch or an augmented reality exploration as a way for students to learn more about things.  After they've checked the video or AR activity, students are given a situation and one math question to answer.  If they get it right, they are told great job but if they miss it, they are told they didn't come up with the proper answer and to try again.  

From a math teacher's perspective, I think I'd ask students to provide their thinking or work as they answer each question.  I would also find additional information to create some sort of sheet to go with this.  For instance, when a student finishes watching the video on the enlisted they are told there are 180 enlisted men and asked how many will occupy a top bunk?  I'd want to know the actual number of enlisted men at various points during it's time of being a part of the navy.

For the information on the anchor, I'd want to know more about the anchor itself such as it's weight, measurements, etc so I could ask additional questions including if we could fit an anchor inside the classroom because most students have no idea how big it is.  We can give them weights but they don't relate to 30,000 pounds but if we said it weighed the same as 5 cars, that is something they can see.

I think it is important to add a bit more math to what the app provides to give students a better idea of how math is found throughout the whole ship but also include activities where they have to relate things to their real lives so they create a connection.  The galley talks about taking 9 pounds of flour to make enough pancakes to feed 100 people and asks how many pounds will be needed to feed 4000 people. I see taking this a couple of steps farther by having students figure out how many pounds of flour will be needed to feed 4000 people pancakes once a week for a month.  Then ask them how many 50 pound bags of flour would need to be ordered for one month, two months, or three months.  Once they have an answer, find out the 3 dimensional measurements of a 50 pound sack of flour and have students calculate the volume the sack is and then ask how much space will the sacks take for a three month voyage.  

By extending the basic questions, it gives students a better feel for what the ship is like.  It could be made into a project where the student is in charge of ordering supplies for a one month voyage.  So in addition to the amount of supplies, students can calculate the total cost of the order.  A real life based project.  Let me know what you think, I'd love to hear.  Have a great day.

Visualizing Combining Like Terms

We have been told to provide visual representations for everything we teach in math.  Unfortunately, the visual of us use tends to be pointing out the squared or cubed associated with the variable.  We assume students know the difference between them but if that is the only way they've seen the terms, they wouldn't know how to visualize them. Without a visualization, they won't see the differences between a squared or a plain x or a constant.  

I took a class this semester on teaching math either remotely or via a hybrid model.  In it she said graphs work as visualizations but what do you use to show the differences between x^2 and x's visually.  I had to think about it but realized I could have students use either algebra tiles or jam board to help them visualize why you cannot combine the two. 

I looked at Jamboard and it is possible to create the large squares, rectangles, or small squares in different colors so they have one color for positive and another for negative.  It would require setting up ahead of time and making copies for each student.  

Jamboard allows students to move pieces around so they can group the like terms and then count the final total for each group.  This can be done by students via distance learning as easily as ones in class.

On the other hand, an easier solution is to use an Algebra Tiles app or online version so students can make as many x^2 as they need in two different colors so they can easily identify positive and negative terms.  They can make as many of each as is needed and move them around  to work out problems.

In addition, this type of visualization can help connect the dots on why the terms in the second part has to change signs for each term following the subtraction sign.  It supports the explanation of subtracting from the original.  

I realize that both of the suggest apps rely on having a mobile device and possibly the internet but what about students who only have a phone with limited data or no internet access?  How do we allow them to do the same exploration as those who have the devices.  I don't have Algebra Tiles as part of my classroom supplies and if I did, I don't think I'd be allowed to loan them out.  Fortunately, there are templets available on the internet.  These can be used to copy tiles onto colored paper to make a physically based set.  There are templets here or here.

Since many teachers are both teaching virtually and have students who do not have internet or computer access, we may have to provide some sort of manipulative with instructions so that we meet the needs of all our students.  This version might require us to send home Algebra Tiles, a mat for them to work on and directions both written and visual.  If a student has a computer at home without internet, the teacher can send a thumb drive home with the video.  If the student does not have a computer, one can create a series of photos, print them out, and send them home.

I have to create two packets worth of work for the first two weeks after Christmas break and I have to figure out how to create the support materials since at least half my students do not have internet access so I'm figuring out how to do it.  Let me know what you think, I'd love to hear.  Have a great day.

Pearl Harbor + Math.

 

On December 7, 1941, the Japanese bombed Pearl Harbor, sinking the Arizona and damaging several more ships causing so many to die.  In addition to Pearl Harbor, a couple of other military places were hit, in addition to quite a few civilians.  This is one of those topics that can be used to create a cross curricular unit connecting history and math.

Many years ago, I helped create a cross curricular unit with math, science, history, and social studies about Pearl Harbor.  Although math can be one of the harder  topics to find this type of activity.

I created three units for students to work through associated with the bombing of Pearl Harbor and the battle of Dunkirk.  I admit that it took a lot of research on my part to find all the information but once I had the units finished, it was worth it.

I'll start with my unit on the battle of Dunkirk in which a flotilla of mismatched ships and boats managed to move over 300,000 people from the continent to the United Kingdom within a short time.  I researched the length of the route, the number of people moved each day, and the type of ships and boats used to evacuate the military after I showed them a video clip on the event.  Students looked at maps, numbers, to see the size of the operation.

For the mathematical part, I asked students how many boats they could scrounge from around the village, and then how many total if we included the next closest village.  In the process they had to figure out how many people could be moved per load.  I had them assume one round trip per hour.  They had to calculate the number of hours it would take people to move the same number of people and then convert the hours into days, and weeks. At the end of this activity, students were amazed at the results and impressed.

I also created a unit on Japanese mini subs, the type that snuck into Pearl Harbor during the attack.  I had to research to find the type of submarine that carried the mini sub from Japan to Hawaii, it's size, the speed of the "mother" submarine and mini subs, etc.  Students used the speed of the mini sub to calculate how long before the bombing they had to leave the mother sub to get to Pearl Harbor, the number of cubic feet the pilots had to fit in as they travelled, and the speed of the mother submarine to go from Japan to Hawaii to place the mini subs in position.  

I used this exercise as a way for students to understand what the Japanese commanders had to consider as they planned the attack.  I admit, they probably used subs closer to Hawaii but I wanted them to see what went into planning something of this size and distance. 

The final activity had to do with the balloon bombs the Japanese released into the air.  Some of these travelled all the way from Japan all the way the states like Oregon, or Washington.  Research is a wonderful thing because it allowed me to find the size of the balloons, the amount of sand used as ballast, the distances from where the balloons were launched to where they landed, the speed of the jet stream and the amount of bomb materials included.  

I included a map of the Pacific Ocean for students to mark down the places bombs landed and draw lines from the place of launch in Japan to the bombs.  They marked down the distances for each one.  Once they had all of this done, they needed to calculate how many hours it would take as a minimum for the balloons to travel to each place.  In addition, they needed to calculate how much paper was needed to create the balloons which required them to calculate the surface area of a sphere and use the results.  They also needed to calculate the total amount of sand needed for all the balloons.  Again it showed students what the Japanese had to calculate before they could even launch the balloons.

I could just as easily have researched the trajectories involved in guns firing from on board a ship, or used trig to determine how thick the steel should be on a ship to prevent ammo from penetrating, the approach of a plane to an aircraft carrier or angle of take off from the deck of a ship.  So many possibilities.  I admit, it will take quite a bit of research to find the information but it is worth it because it makes some of these events more real.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If a dolphin normally swims at 6 miles per hour but has a top speed of 20 miles per hour, what is the percent increase from normal speed to top speed?

Warm-up

 

Give your age without using years.  For instance, I am one quarter, one dime, one nickel and three pennies.  My friend will tell you her age is equal to four 12 foot ladders.  How old are you?

Near and Far Transfer..

I love researching new topics because I’m often lead to new ideas and thoughts.  During my research on the two types of knowledge, I ran across some new information on transference of learning.  Information that helps me understand why students often time have difficulty in taking the information they learned in one situation and transferring it to another situation.

I just discovered that transfer learning can be divided into two types.  The first is near transfer which involves skills and knowledge which are applied the same way every time while the second, far transfer, is the ability to transfer those skills and knowledge to other situations.  An example of these would be for near, the students are taught to calculate percentages and can apply them to any problem in the book. This is an example of near transfer because students can apply what they've learned to the same type of problems.  To be considered far transfer, students would have to be able to go to the store and calculate the percentage taken off a jacket based on the original price and the new price.  

This is where so many math teachers get frustrated.  They see students can do the problems and can pass a test but when they are asked to use the same knowledge in a more real world situation or to a situation that is not identical to the context of what they learned, students can't and we wonder what happened.  Now we know.  They've mastered the material as a near transfer but not a far transfer.

One thing that's been noticed is that real world application is often times more complex than the problems students have solved in the classroom.  I've heard it said that real life is so much messier because the answers are not as neat and tidy as most problems students experience in class.  Consequently, it is possible that the problems taught in class are too simplistic and this can make it much more difficult for students to perform far transfer.

Fortunately, there are some things teachers can do to help students learn to apply knowledge via far transfer rather than remaining in near transfer.  The first is to engage students in working with real world applications of what they've learned.  Unfortunately, this can be much harder in Math because many teachers, including myself, do not always know how certain topics and concepts are used in real life.  Next, it is important to help connect what students are currently learning with what they have learned in the past.  For instance, we teach students to solve equations and inequalities the same way so as we teach inequalities, we can talk about solving regular equations.

 

If students are given extensive practice for routine skills such as addition, subtraction, multiplication, and division so students can perform them quickly and accurately.  Being fluent and comfortable, also contributes to a better near transfer because students can focus on the concept rather than struggling with the arithmetic.  As far as assigning problems for a specific skill or topic, it is important to assign a variety of problems because the more varied contexts students see, the easier it is for them to transfer the knowledge from near to far learning.  This means that instead of having all problems with the equal sign on the right, make sure it is appears on the right side.

Furthermore, it is important to for the instructor to point out the underlying principles are used in different situations so students see how to use them in a variety of situations.  This makes it easier for students to utilize far transfer because they've been learning to recognize the underlying principals and then apply them.  Finally, students need to be taught to reflect on their own thinking so they can improve their learning.  

It has been suggested that all learning goals be written in two parts.  The first part covers the procedural objective which is the part where students learn the steps such as in learning to solve two step equations.  The second is the declarative part which involve the conceptional knowledge or how to apply it to a variety of situations or contexts.  The second part is so much harder because I've never thought of doing it.  Let me know what you think, I'd love to hear. 

Inert vs Generative Knowledge.

When we teach math, we usually seem to want to teach procedures without taking time to focus on having them see how everything relates.  The other day, when I read up on curriculum, I ran across a reference to inert and generative knowledge.  I honestly don't  remember hearing about either.  Knowledge was knowledge.

Briefly inert knowledge is knowledge that is not used while generative knowledge is used to solve a problem.  Remember back in elementary when you had to memorize all the state capitals and never did anything more with those?  That is inert knowledge but if you used the information to write to the governor of each state, then it becomes generative knowledge.

There are three types of inert knowledge.  The first is that the knowledge is there but not accessed while the second states there is a problem with the structure of the knowledge.  The knowledge is in a form that cannot be applied and the final is there is an issue with the situational usage of the knowledge.

In math, we often teach students process used to solve  various types of problems and we give lots of practice problems but we do not provide situations that require students to apply the processes to solve problems.  In fact, most real world problems found in textbooks are neat and only require students to apply the math learned in the section.  

Generative knowledge is often referred to as generative learning.  In generative learning, it is believed a student is not going to learn the material as well as when they are able to construct meaning by generating relationships between what is learned and it's usage. In other words, they are generating understanding.  If teachers do know help students generate their own understanding for each new topic or section, then they will know how to use it to solve problems, otherwise each topic or section will be treated as in isolated skill that cannot be applied to problems unless the student is taught to explicitly apply it.

This may explain why students seem to know what they are doing but are unable to apply it when they take the state test, or a district test.  This is because it remains as inert knowledge rather than being moved to generative knowledge.  Fortunately, there are ways to help students to this.  One way is to ask questions of students that make them look at similarities and differences between processes or topics.  Another way would be create discourse that encourages discussion, debating, or generalization.

In addition, it is important to create situations where students can apply the knowledge they are learning so it is no longer inert.  For instance, when I teach solving one and two step equations, I take time to show how the same equation written in a general form can be graphed as a line and how the solution for x and the answer (y) is a point on the line.  In other words take the problem 2x + 3 = 7.  We solve the equation to find x = 2. I teach that x = 2 and y = 7 and that point is on the general line 2x + 3 = y.  

I try to relate topics or concepts to things students have had before but I need to provide more activities and discussions to help students create their own understanding rather than trying to do it for them. It is hard sometimes because they arrive in high school not having had a lot of experience creating their own meaning.   Let me know what you think, I'd love to hear.  Have a great day.