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.

The Different Types Of Curriculum?

 

The curriculum is something all teachers have to face.  It is the list of what we are supposed to teach in a year.  It used to be a huge book filled with pages of scope and sequence and material that seemed to get added every time it was redesigned but nothing ever seemed to be removed.  Eventually, they looked like door stops.

Recently, it has been replaced with "Common Core Standards".  The ones used in Alaska are broken down for grades K to 8 but for high school, all the math standards are mixed together and teachers have to separate everything out.  

This year, the math curriculum department went through the textbook series, deciding what should be taught, what can be skipped, and what can be lightly touched upon.  Although it is cut back, there are still areas, I need to include several suggested omitted topics because they are often the foundation of the skills I need to teach.

In addition, about one third of the students never finished Algebra I and they are lacking certain skills needed for Algebra II so I have to teach them.  For those who completed Algebra I, the year before, I have to review certain skills so students are able to do the work. I also have to provide more scaffolding to help students get through the class.

When I talk with teachers who graduated in the past few years, they always refer to curriculum as the textbook the school uses rather than what should be taught. I looked up curriculum to find out when it became associated with textbook series rather than a guide on what should be taught.  It appears there are multiple types of curriculum which may explain the confusion.  

First there is the recommended curriculum which covers the topics that experts feel should be covered.  Then there is the written curriculum found in most state documents as "standards" or in the school website, specifying what should be taught. In addition, there is a supported curriculum done via textbook series, software, and multimedia materials.  Then there is the tested curriculum or what the state, companies, schools, and teachers create and test students on.  Of course, there is the taught curriculum which is what teachers actually manage to teach regardless of the written curriculum or pacing guides.  Finally, is the learned curriculum which is what the students have learned and this is really the most important. 

The question then becomes how do these all relate to what and how topics are taught in the classroom.  Although the recommended curriculum is what experts believe should be taught, it has less influence on the written curriculum and even less on the classroom teacher because they have to take into account their students, what has worked in the past, and what appears on district and state tests. In fact, the material on state and district tests seems to have the most influence on what is taught in the classroom.  

The supported curriculum has more influence on elementary teachers because they have to teach multiple subjects and the textbook forms the basis of their content knowledge.  Furthermore, there is always a gab between the taught curriculum and the learned curriculum due to short attention spans, a lack of motivation, a failure to monitor student progress, and a failure to make the topic meaningful and challenging.  

To have a high quality curriculum it should focus on a smaller number of topics in greater detail which is the opposite to the commonly accepted pacing guide which covers a bunch of topics with just a glance.  It should also have students using various learning strategies to solve problems. Students need to acquire both essential skills and knowledge of a topic so they understand it better.  In addition, the curriculum has to be set up so it meets students individual differences. Make sure the classes are multi- year, multi-level sequential courses so as to build upon previous knowledge.  

Take time to focus on learning a smaller number of essential curriculum objectives while maintaining an emphasis on what has been learned.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If it takes 14 ounces of cranberries to make a small 10 ounce jar of cranberry jam, how many jars of jam will 150 pounds of cranberries make?

Warm-up

If the average serving of turkey for each person is .25 pounds, how many people will a 18 pound of turkey feed?

Love The Book "Reach Them All"

I seldom get around to reviewing books but this is one I think could easily solve some of the issues we face as teachers.  One big issue is how to work with students who are missing certain fundamental  skills which make it harder for them to do the expected work.

Most of the time, the recommendation is to work on filling the gap while teaching the current unit rather than giving them the work at the level they test at. Unfortunately, I've never seen a plan which provided a reasonable way to accomplish it. Thi s is where this book by Chris Skierski comes in.

He suggests teachers, including high school teachers use Learning Stations to accomplish this.  The idea is the teacher determines which basic skills are needed to do well with the current topic.  For instance, if you are teaching students how to solve one and two step equations, they should be able to add, subtract, multiply, or divide integers both positive and negative before the lesson begins.

The process involves a pretest, learning stations, and a path of specific requirements to move on.  Students take a pretest to determine if they have all the necessary skills to begin solving one and two step equations or if they are missing some skills.  The results of the pretest determine which learning station the student begins with.  Each learning station is set up with a video on the topic and students are expected to at least write down the examples given on the videos.  Then they practice on-line at IXL, Khan Academy, or other location.  This is followed by a work sheet and then they finish the station by taking a quiz.

If the student does not pass the quiz, they are expected to redo the material with different videos, practice, worksheets and quizzes.  If they pass, they move on to the next learning station until they've completed every skill needed for solving one and two step equations and the lessons on the topic.  In addition, as students complete the skills, there is a small celebration held acknowledging their success.

The book includes all the forms, details, and information needed to create this in your room. Furthermore, students are not expected to work at learning stations for the whole period so the process is broken down into smaller chunks to be completed each day. For instance, the student should do the video and notes on the first day, the online practice on the second day, the worksheet on the third day and on the final day, the student takes the quiz.  In addition, there are some enrichment or extensions included for students who have most of the skills but just need a brush up.

I have been looking for how to do learning centers effectively in high school and this is the first book that offered solid information.  Usually the information I've found on learning centers is geared for elementary or just tells you what should be there in a vague way.  This book provides concrete examples so I can set up learning stations for all the topics I need to cover.

My next step is to sit down and figure out how I'm going to incorporate learning stations into my classroom. I've realized that I can combine topics in the book so they line up with basic skills such as combining like terms when I address adding and subtracting polynomials and binomial or trinomial multiplication for multiplying polynomials.

In addition to the book, Chris has a great website with some really helpful resources and classes. At least one class is free and I've signed up for it and need to get started. Let me know what you think, I'd love to hear.  Have a great day.

Happy Thanksgiving

Pacing Guides Are Unrealistic!

 

I am trying to figure out why districts have the idea that the pacing guides provided by math textbooks are realistic.  I have a pacing guide for every textbook I teach from and I can tell you the one I have is totally unrealistic and do not meet the needs of my students.  

It seldom allows students more than 2.5 days on any particular section but most are between a day and a day and a half. The lessons begin with an exploration, vocabulary, notes, practice and then the assignment.  Most of the explorations take my students at least 20 to 30 minutes instead of the allotted 10 minutes.

Unfortunately, the author's idea of differentiation is to assign different problems depending on a person's level of performance.  Every lesson is the same and none of the lessons are designed to really help students who are behind.  The year before I arrived, the district insisted teachers follow the pacing guide as written. If  a teacher got behind, they had to meet with the principal to determine what could be done so they got back on pace. Last year, they didn't push as much and when the pandemic hit, everything fled from their minds.  

I've known for a very long time that when we follow any pacing guide, we are just forcing them through a class with no regard to their actual learning.  The lessons are not designed to take time out to help students work on weaknesses, and really learn.  It doesn't really follow the practice of breaking information down into small, understandable chunks.  If you have students who are classified as English Language Learners (ELL), the pacing doesn't give them a chance to work on vocabulary or learning the language of mathematics.

How can we expect students to learn and progress if we do not include time for meaningful exploration, a chance to develop visual representations for various concepts so we can take students from concrete to abstract.  It's like the pacing is designed to get students through the whole book rather than learning the concepts.  It makes students feel as if they just need to make it through lessons, put the "right" answer down without transference of knowledge.  This may be why students often see each problem as something new.

I think it is also responsible for students not seeing the connections between basic topics and their applications to various situations.  That may be due to the pacing because it doesn't give students a chance to develop relations between one topic to another.  In addition, most lessons still rely on material and methods from when I was in school. It's almost as if everyone is brainwashed into believing one has to teach students everything in the book. It just seems to me that if we expect students to learn, we need to either cut significant amounts of material , or we have to throw out the pacing guide.

Rather than following a pacing guide, I would rather have a list of the most important material students should learn, and the time to present it in small chunks with additional time to provide differentiation, scaffolding, and a chance to let students work on any missing skills at the same time.  I've been at schools where the mantra "We are here for the kids" resonated around the school but we still followed the pacing guides rather than looking at where the students were and deciding ways to help them gain their missing skills.

On Wednesday, I want to share a few ideas from a book I am reading that makes so much sense.  It is totally in opposition to following a pacing guide yet provides a proper way to help students fill gaps and move forward.  Let me know what you think, I'd love to hear.  Have a great day.

Thanksgiving Math Activities

 

It is that time of year again, the Wednesday right before students are off to enjoy Thanksgiving and a four day weekend.  We know they aren't interested in doing a regular lesson so why not plan an activity that uses math to address some aspect of Thanksgiving.

One site that can be counted on to have Thanksgiving themed math activities is Yummy Math.  If you check the site, they have an activity on cranberries.  In Alaska, the cranberries we grow are quite low and close to the ground since they are classified as low bush cranberries.

The cranberries the activity cover those grown commercially for juice, sauce, jelly, and the fresh ones sold at the grocery store.  The activity looks at why they float, how much each weighs and how many make a pound, etc after watching a video on harvesting cranberries.  Students are expected to estimate, predict, calculate, and use problem solving skills to complete the worksheet. 

In addition, to looking at cranberries, there is an activity on cooking your turkey, making mashed potatoes from so many pounds of potatoes and making pumpkin pie for what people eat.  There are also a couple activities on football.  One activities have students interpret information about NFL teams on an infographic while the other activity has students creating a graph from data on fourth quarter decisions  whether to punt or try a field goal.

There are also several activities designed to look at consumer spending at this time of year and the savings people find during the Black Friday sales.  In addition, this site has two activities relating to Macy's Thanksgiving day parade but they have the cravat that these are older and do not take into account the changes Macy made for this year.  The final couple of activities look at the distance someone ran and building structures out of cans.

On the other hand, Math-aids has some four quadrant coordinate plane pictures available so students can practice plotting coordinate points and end up with Thanksgiving themed pictures. There are two turkey pictures, one pilgrim hat, and a pumpkin with a pilgrim hat.  Once students finish graphing the picture, they can color in the final product. 

Finally, Wallethub has a great Thanksgiving infographic that includes information such as the average cost for a 10 person dinner, or how many total calories consumed by Americans on Thanksgiving day. It lists people favorite dishes, the number of hours a male has to work to work off a meal, the number of total turkeys killed, the number of questions answered by the Butterball hot line and do many other interesting facts.

It makes an awesome what do you notice?, what do you wonder? and what type of mathy questions could be asked about this infographics.  In addition, it would be easy to have students take some of the information and turn it into pie charts, bar graphs, and other graphical representations.  Let me know what you think,  I'd love to hear.  Have a great day.

Warm-up

If  2.75 cups of dried rose petals are in one ounce, how many cups are needed to make a pound of rose petals?

Warm-up

If there are 14 cups of dried lavender buds in one pound, how many cups of lavender buds are in an ounce?

Sunrise, Sunset Data = Sinusoidal Waves.

 

Want to get students involved in a project which will help them model real life data using a trigonometric regression?  Set them up to learn more about the increase and decrease of sunlight throughout the year in various locations around the world.  Arrange things so they do the whole project from the beginning by collecting data, to using a sin regression to find the equation to identifying all the parts of the transformational sin formula.

The first thing to do is to assign each student a city such as Honolulu, Hawaii,  Reykjavik, Iceland, Oslo Norway, Brownsville, Texas, Rome, Italy from all over the world. The choices should range from far north, to around the equator, northern hemisphere and Southern Hemisphere so students have a chance to see how the sin wave changes according to it's location on the earth.  This site has sunrise and sunset information from most major cities around the world. 

Students should write down the times of sunrise and sunset for same day every month.  When I tried out this activity, I chose the 15th of each month because it was the middle which I felt was an average.  After recording the data, students should subtract sunrise from sunset to get the total length of daylight.  The next step is to create a state plot of the data which can be done by hand or on a calculator.  

I've seen the data entered in two different ways.  The first way is to count the months so January is 1, February is 2, all the way to December being 12 and the second piece or L2 is the day length, or you can count the days themselves so Janury 15 is day 15 which February 15 is day 45, all the way to December. This is a great opportunity to discuss which way would be better.

This is a great point to have groups of students place several plots onto one graph so students can see the similarities and differences based on the different locations.  The cities should be grouped so one is located fairly close to the equator, another as far north as possible and two others somewhere in-between.  It is interesting to see how the data curves if it is from say Oslo, Norway compared to Honolulu, Hawaii. The one from Honolulu will have a smaller stretch than the one from Oslo.

Once all the data is entered and the stat plot is completed, students will be able to see that the sine waves will provide the best fit so this is a great opportunity to teach students how to carry out a sine regression to find the an equation for the data so it its the f(x) = asin(bx + c) + d general formula.  In addition, one can show how the a, b, c, and d are calculated individually so students see how each part is calculated.  At the end, they can create a written report comparing and contrasting their city with others.

One of the easiest things to find first is D because one just has to add the maximum and minimum hours of daylight together and divide by 2.  That would be the vertical shift and center line.  To find A which is the distance form the maximum to the center line one just has to subtract the maximum -  D to find A. In order to find B it's just 2 pi/the period such as 365.25 for the year.  Once students have A, B, and D, they can calculate C.

This exercise provides a wonderful real life application of sin waves where students learn more about how amplitude is found, how b is pretty much standard with 2pi/period, and the horizontal and vertical shifts.  Real data, real calculations and real modeling.  Let me know what you think, I'd love to hear.  Have a great day. 

Real Life Data = Piecewise Functions

Another talk I attended discussed the use of real life data and then figuring out an equation using one of the regressions for line of best fit.  Unfortunately most text books focus only on using linear regression to find an equation for the line of best fit for a scatter plot.  

This talk was so interesting because the presenter used more than linear regression to find an equation for the data points.  This person provided real life data from the CDC on the number of opioid deaths over the years.  

Once he had the data inputed and the stat plot up on the screen, he began using various regressions to see which produced a line that fit the data as well as possible.  It was obvious the standard y = ax + b created a line that had an r^2 indicating it wasn't that accurate.  Furthermore, the end of the data showed a steeper curve indicating the equation needed might be an exponential one. 

I loved the way he showed us how equations created by quadratic, cubic, quintic regression matched the data.  This lead us to seeing how different parts of each equation matched up with the curve of the data.  It showed a real visual reason for using piecewise functions to end up with a series of functions to match the data.  In the past I have taught piecewise functions but I never had a context for why one would use it but now I understand a good application of it.

After watching the presentation, I realized that I need to include this type of information when teaching either linear regression or piecewise functions.  My pre-calc students are taught linear regression in the first chapter without a proper context so I think I'll expand that lesson the include other types of regression applied to the same data so students understand the line of best fit might be made up of sections of a variety of equations.

The presenter used the TI-84 to show how to do each regression and the piecewise.  He used a newer one that had the piecewise function choice.  The ones I have do not have that choice and as much as I like the TI-84, Desmos is easier to read for discontinuities, find the values for intersections, and other pieces of information.  I actually have students use both so they have the ability to use both applications. 

This gives students the experience to use a variety of tools depending on the circumstances.  They have to be able to use both because they may not have access to a handheld calculator but they do to the other.  It is important they exposed to a variety of tools.  It appears that Geogebra can also be used to create a variety of regressions. 

The gentleman also had data for the coronavirus and a few other things which makes it easier to start teaching the topic with something more relatable for my students. Furthermore, I don't think I've done any regression other than linear regression since college and even then it was by hand because they wanted us to know how it worked.

So next time I teach linear regression I am going to use this information and the different programs to create a piece wise equations. I'll be looking for some nice reliable data for other activities.  Let me know what you think, I'd love to hear.  Have a great day.

Circle Packing = Social Distances

I just finished attending the Virtual NCTM meeting this year and I loved being able to attend talks without leaving home.  In addition, it is great I'll have access to recordings because I can see the talks that I missed due to all being at the same time.  One of the talks, a short 30 minute one, looked at circle and sphere packing and it's applications to the modern world.

The one thing the speaker said throughout the talk is that once people figure out the math, they never know all of the idea's possible applications at the time.  He began with the circle inscribed inside the square.  The square was say one unit by one unit while the square had a radius of 1/2. 

He showed how the area of a circle is about 78% of the area of the square.  I gather this is an old problem whose new application only became apparent during this pandemic with social distancing.

With today's 6 foot distancing mandates, offices often have to figure out how to arrange workers so companies can have as many workers as possible while still maintaining social distancing.  I used the basic idea when I arranged my classroom for my biggest class while trying to maintain the appropriate distances.  

The specific idea being used is called circle packing which is where circles are arranged within a certain area so that many of the sides are tangent to each other but do not overlap at all.  Most people think of packing circles within a square area by doing rows upon rows, so everything lines up such as in the picture of flowers.  

This is not the best arrangement for circles because there is still quite a lot of space available between the circles.  This is one of the usual arrangement teachers choose for their classroom, rows of seats all neat and orderly.  As stated earlier, this is not the most efficient arrangement.

After a lot of exploration, mathematicians discovered the best arrangement is actually hexagonal with a density of almost 0.91.  I didn't realize the hexagonal arrangement was the most efficient when I arranged my classroom. I just eyed things and put the chairs into this orientation because it was the only way I could figure how to arrange all the chairs to maintain distancing.  Although it seems that it might be a good idea to pack circles in side a circle, it really is not the most efficient.  

In addition, the idea of circle packing can be extended to sphere packing which is the idea that one can arrange spheres in the most effective arrangement.  This problem was first proposed in 1611 and Kepler came up with a solution but it was until around 1998 that it was proven the best arrangement is just the way they stack oranges in a pyramidal shape at the grocery store. Others researched the best arrangement up to 24 dimensions and found one that works.  Sphere packing is used currently for data transmission and error correcting code. So we have something first proposed in 1611 whose applications are extremely important in today's society.

I found this extremely interesting.  Math first proposed centuries ago that is now playing a huge part of our daily lives.  Let me now what you think, I'd love to hear.  Have a great day.

Warm-up

If one candidate is leading by 0.26% and that equals 27,000 votes, how many votes would he be leading by if he were ahead by 1% of the votes?

Warm-up

If 130 secret service people have tested positive or are in quarantine out of a force of 1300, what percentage cannot work?