Math for New Year's Eve!

Today is December 31st, also known as New Year's Eve, a time when we say good bye to the old year and hello to the next year.  Its a time of partying, of drinking, of eating herring or beans or other family tradition.  It also provides a wonderful set of activities we can integrate into the math classroom.

Yummy Math has a lovely activity focused on the Times Square celebration.  It provides a time line of the Times Square ball from 1907 to the present.  The activity asks students "What they wonder?" and "What questions they have?".

When I saw the data, my first thought was to graph it by diameter, weight, or number of bulbs used in the ball.  From a bit of research I've done, the pole is 141 feet and the ball is geared to drop in 60 seconds even back in 1907.

However, it is possible to research the number of people who go to watch the ball drop and create a ratio of number in the square vs the number of citizens in New York City at the same time.  For instance, I know there were 200,000 who turned out of a population of approximately 6 million people.  Since most of my students have not been out of Alaska, they have no idea what Times Square looks like.  I'll be honest, I am only familiar with the picture they show but have no idea what it looks like.

This is where Google Earth comes in handy.  It is possible to pull up the area where the Times Square ball is dropped so students can explore it using street view. They can also look at the over head view to see what the area looks like and get a better idea of the restrictions placed on the number of people who can view the ball dropping.

Its possible to find the numbers of people who visited Times Square over various years and find approximate populations for New York City at the same time to find a ratio.  It is possible to also determine the density for the area based on a bit of research.

Although viewing the dropping of the ball doesn't cost anything, it requires people start assembling in the area around the ball by 3 PM but if you want a prime view with dinner in a warm area, then people buy tickets for hotel activities.  This can easily be researched on line.  I did a quick search and discovered tickets are available for $999 to about $1400 per person depending on the hotel.

The costs of a party at a hotel provides possibilities for creating infographics, charts comparing what you get when you purchase a ticket to watch the ball from the warmth of a penthouse event.

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

Warm-up

Warm-up

5 Ways to Engage Students

 One of the hardest things I have to deal with, is that of motivating students who see little use for school or for math.  Their parents and grandparents may never have finished high school, yet are doing well within the context of living in an isolated village.

Unfortunately about 52% of registered students are absent at any one time which can make it difficult for students to learn the material because they end up missing foundational pieces.  I believe this is why many of my incoming freshmen have significant gaps in their knowledge of mathematics.  Of the 48% who make it to school another 10 to 15 percent either sleep or do absolutely nothing because they either don't have the skills or have exhibited this behavior in previous grades.

One thing that is going to help students during the second semester is the iPads that were finally issued to my classroom.  I've been waiting for them to be distributed since August and I only got them at the beginning of December about 1.5 weeks before holidays.  Now I can post thing on Google Classroom so students can keep up with material when they are absent or are traveling.

So what are some ways that might help students become more motivated.  Motivated to come to class.  Motivated to keep up with work.  Motivated to even try.  Some of the suggestions I've read will not make a difference to motivating my students but some will.

1.  Relate the mathematics they are learning to the real world and to the world they live in.  Since my students live in a village accessible by air, trying to do problems with trains or automobiles leaving two different cities makes no sense but if I rewrite those problems using places they know and vehicles such as ATV's or Snow Machines, they can then relate to the problems.

I know one teacher who taught her students the words and pictures for items they'd likely encounter on a standardized test and equated the items to ones they understood.  For instance, many students fish using a piece of board and fishing line rather than a rod and reel so she taught them the rod and reel while relating it to their more traditional fishing methods.

2. Allow students to divide up in small groups of three to four people who will work on the problem assigned by the teacher.  The first group with the correct answer and the group with the highest number of points at the end of the period receive a prize. Another activity along this line is to provide a menu where students choose certain problems to do.  Students do well with choice so they can work problems they feel better able to do.

3. Use props to grab their attention or to illustrate formulas or concepts being taught.  Props can help students learn the material better.  If they are engaged, they are learning.

4. Assign problems with more than one answer.  Let the students know there is more than one answer and challenge them to find as many as they can.  Or chose a problem with more than one way of ding it and challenge students to find the multiple ways.

5.  Finally set up an environment which allows students to feel safe.  If they don't feel safe, they won't try and will set themselves up for failure but if they feel safe, they will try.  Its a matter of balance.

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

Writing Tiered Lesson Plans

  Now for information to write tiered lesson plans to go with the tiered assignments from yesterday.  It is something I need to really look at since I discovered too many of my students test at grades 4 and 5.

There are eight steps to look at each time the teacher writes a tiered lesson plan.  Understand that depending on the criteria used to divide students into groups may require the teacher to adjust groups as needed.

So look at these things every time you write a tiered lesson plan.

1.  Identify the grade level and subject for the lesson.

2.  Look at the standard you will be addressing in this lesson.  Don't wait to the end, do it now.

3. Identify the key concept or idea.  Ask yourself what is the big idea and what you want the students to know at the end of the lesson.

4. Decide if students have the necessary background knowledge or do you have to provide it?  Is scaffolding necessary?

5. Determine if content (what you want them to learn), process(how students make sense out of content) or product(outcome of the lesson).  It is recommended to only do one if you are just beginning.

6. Determine what type of grouping you should do.  One way is to divide students based upon pretest assessment.

7. Divide the students into groups.  It is recommended one use only three groups because if you go for more, it becomes more difficult.  The standard divisions could be into below level, on level, and above level.

8. Develop the assessment associated with the lesson which could be formative, summative, or both.

So now you know the basics for writing tiered lesson plans with tiered assignments.  Let me know what you think, I'd love to hear.

Creating Tiered Assignments

 Many of us work with students who see any math problem and automatically see it as "too hard".  They won't even try because they feel they can't do this.  One way around it, is to use tiered assignments because students have a choice and can choose those problems they think they can do.

Tiered assignments are a form of differentiated instructions that can be as simple as letting students choose problems based on their ability to having a much more complex structure.

Lets look at how to create tiered assignments for your classroom so you meet the needs of everyone, not just one group.  In order to create a tiered assignment there are some things you have to think about first.

The reasons for using tiered assignments include having students begin where they are, reinforces or extends material, allows students to have work which they do not perceive as too hard.  It promotes success.  

Before writing the assignment, you need to decide what part can be tiered?  Is it content? or process? or product?  Is this material being used to check readiness, interest, or test scores?  Then when you begin writing the actual assignment make sure it follows these suggested guidelines.

1. The task needs to cover the key concepts or generalization that is essential to the topic.

2. Use a variety of materials of differing levels of difficulty.

3. Adjust the task by complexity, number of steps, and independence.

4.  Let students know the grading criteria.

A good way to create the actual lesson is to answer these three questions.

1. What is the range of learning abilities?
2. What should students know, understand, and be able to do at the end of the lesson?
3. How will you hook students? What will they be able to do at the end of the lesson?
4.  Prepare three levels of the same assignment from easy to more challenging.

This process can be used for preparing lessons taught in the class and for daily assignments.  Tomorrow, we'll look at writing tiered lessons in detail to go with the tiered assignments.

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

Merry Christmas

The Math of Christmas Eve.

  I found it!  I found an article where someone looked at how many children,Santa has to visit on Christmas Eve so each and everyone gets a present.

The author explained where he got his numbers and what criteria he used for selecting numbers.  For instance he looked for the number of christian children aged fourteen and under.  he found there were 526,000,000 who celebrate Christmas on December 25th.

So if one assumes you have 24 time zones Santa has to travel through beginning late on the 24th so the gifts are there for the children in the morning, the author calculated Santa has to deliver 22 million presents each hour.   This breaks down to 365,000 children per minute or 6,100 children per second.

In addition, there are some wonderful graphs for students to practice reading.  One of the graphs breaks down the population according to time zones while the other looks at areas with the 22 million children in a time zone.  The two graphs compliment each other

Then towards the bottom, there is a short slide show discussing which country or countries are the most populous in each time zone, its population of children, and how long it should take Santa to deliver the presents.

Furthermore, at the bottom of the article, the author takes time to list issues which might effect the actual numbers such as not all Christians celebrate Christmas such as Jehovah's Witnesses and some non-Christians do celebrate it so that might change the numbers a bit.

There are also issues where certain countries cover several time zones which makes calculating things a bit tougher.  I really liked the article because the author really takes time to break it all down.  It is well written and worth using in class because its something you can assign to students to read and if you provide an accompanying worksheet with questions for students to answer.

Check it out.  Let me know what you think, I'd love to hear.  Have a great Christmas.

Alaskan Christmas

Found in the Anchorage Alaska airport by Security.  Warm-ups return after the New Year.

Thoughts

I have seen this guy entertaining travelers who are making their way through Sea-Tac airport.  I wonder how many have seen him or stopped to listen.  Sorry, its not a great shot but he was on the move and didn't stop.  He's a one man marching band.

Learned Something New

Yesterday was one of those days where our flight was delayed due to one of the more unusual issues.  Usually, we have planes run late due to weather, ice fogs, fog, or something similar but yesterday it was labeled as "maintenance".

I called the airline to discover they hadn't properly put their Navajo Caravans to bed the night before so when the temperature really dropped, the electronics got messed up.

They had to fix systems such as the auto pilot and other things in order to come pick us up.  I knew that if you don't plug your car in during super cold weather, the battery might loose power or the oil might get too thick but I never thought about electric circuits.

In the process of learning this, I realized I have the basis of a great lesson based on temperatures and the relation to the effectiveness of electronics.  I discovered that certain companies set electronics to work between certain temperatures such as Apple which sets their iPhones, etc to work well between 32 and 95 degrees F while Samsung sets their electronics to function best between -4 and 122 degrees F.

I figure I can have students research these tolerances for various electronics from different manufactures to create graphs based on those tolerances.  From the graph, they can provide a written paragraph explaining which manufacturer they recommend.  This is very appropriate for us since temperatures regularly drop to the -20 F range and colder.

In addition, I just discovered that the effective range depends on if it commercial, industrial, or military based with military based having the best spread for temperature range.  I'm also sure with a bit more research they could find graphs showing more of a degree by degree connection with effective operation of the electronics. 

Furthermore, since most of these items operate off of battery power, the project could include a section on the temperature range of effective use for batteries.  In cold weather, we have to wrap the car batteries in a battery blanket so it stays warm over night and the battery does not loose it charge.  For up here, we have to get the heavy duty batteries designed for the low temperatures. 

So many possibilities and my students can actually relate to this because they have to start a ATV or snow machine in cold weather and its a challenge.  Many throw blankets over their machines while others use a hair dryer to warm everything up.

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

Interesting Training

  Yesterday, we had a very interesting training after school.  Just a bit of background first.  The super decided to transform our school from one principal and one assistant to two principals.  He divided the school into elementary with one and middle/high with the other.

So yesterday, the soon to be middle school/high school principal ran a training on what he wants in his lesson plans.

He is the first person in all the time I've taught at this school to do that. He even modeled it which was great.  He wanted to do this so we'd know what to expect when we got back.  So here it is:

1.  Write a simple objective.  It doesn't have to be anything fancy but it has to be able to be assessed so you know if you met the objective. 

2.  This is where you teach the main lesson.  No student should be talking and it shouldn't be longer than say 11 to 15 min at most.

3. This is where you check for understanding through discussion or centers.  You are expected to wander around checking and making sure they are on task.  If they are engaged, let them talk longer.

4.  This is where you have students talk in a think, pair, share, or give each person in a pair a different problem that they do and then teach the other one.  Keep them busy.

5. Only after you know the student knows how to do the work, do you give them the assignment which will be graded. 

6.  The final step is providing closure.  Provide the answer of when and where this will be used in the village or in real life.  Unfortunately, for math it gets a bit harder because we only have a once a week newspaper with few graphs and few sports results.  I know there are two stop signs but no children at play or yield signs.

He also said that we needed to put the lesson plan in the classroom where he can get to it easily and check to see if we are doing Tuesday's lesson on Tuesday rather than Thursday but that tends to happen out here a lot. 

The best news of all is that I finally got my iPads so now I can set up my google classroom, and other things.  The bad thing is they haven't put all the apps I've requested so I don't know if they will do that over holidays.

I'm off, have a great day.

Christmas Program

Today is the Christmas program, one day before holidays.  Picture 300 students grades pre-K to 12th all going on to entertain parents but first the head start children file on, look cute, and leave.  It will take a couple of hours to complete.  I've spent the past two days, offering students a chance to pass so they are eligible to play basketball next semester.  Should be back to normal tomorrow.

The Mother of Cryptology

The modern mother of cryptology is a woman whose achievements were lost in time due to others claiming responsibility for what she did. 

 Elizebeth Smith Friedman is a woman whose name is not one that immediately pops to mind when discussing cryptography or mathematics.  In reality, her husband is much more well known since he helped found the NSA.

Elizebeth was born in 1892, the youngest of 9 children.  Although she received her degree in English lit with strong studies in Latin, Greek, and German, she ended up applying for a library job that resulted in being hired at a private think tank in the late nineteen teens.

At this point in the, this facility had the only national cryptologic laboratory devoted to trying to prove Sir Frances Bacon actually wrote all of Shakespeare's work.  It was here that she met and married William Friedman, her husband and partner.  After a few years, they moved to Washington, D.C. to work for the government.

Although a poet and not a mathematician, she taught herself how to decode secret messages without knowing the key.  Her mind was incredible to crack codes, finding keys, and creating methods that changed the world of cryptography. Furthermore, she helped invent cryptology or  the modern science of secret writing.

One of her first jobs for the Navy and Coast Guard had her breaking codes used by rum runners or those who illegally transported alcohol and other goods during prohibition.  She helped capture criminals and testified at numerous trials which resulted in convictions because she easily explained how she cracked the codes to juries. During this time, she cracked over 20,000 messages whether simple code, transposition, or something more complex.

In 1937, she helped the Canadian government convict an opium dealer by cracking the code based on Mandarin Chinese without knowing the language. Just a few years later, she and her team of code breakers began intercepting messages that were quite similar to the prohibition type messages but were sent by Nazi spies.

Just before World War II, she transferred to the Coordination of Information where she
focused on cracking certain Enigma codes, specifically those based in South America, so messages could be intercepted.  She shared the codes with the FBI, giving them the ability to intercept and translate messages resulting in the South American spy network being shut down.  J. Edger Hoover claimed the FBI did all this on their own and ensured no one knew she had anything to do with it by creating a propaganda film.

She also discovered the letters written by Velvalee Dickinson contained coded information about the moves of ships at Pearl Harbor.  Her work was responsible for Velvalee's conviction in 1944 as a Japanese spy.  Five of her letters were sent to Elizebeth because the letters with their talk about dolls in her doll collection seemed strange.  Velvalee did own a doll shop in New York City where some of the dolls sold for as much as $750 each but upon investigation, it was noted that she had fallen into debt upon her husband's death.  Investigators discovered her ties to the Japanese consulate and the Japanese American Society which was enough to arrest and try her.

Although she had no direct link to beginning the National Security Agency,  she helped create the science underlying their work. It was her husband who helped found it with the Army code breaking unit he founded in the 1930's that was later absorbed into the NSA.

If you are interested in her story, check out the book "The Woman Who Smashed Codes" by Jason Fagone.  Let me know what you think, I'd love to hear.  Have a great day.

Knitting Magazine

  I collect knitting books and magazines for the day I have more time to enjoy myself.  I picked up a magazine that was not the usual one filled with capes, sweaters, and tops.  It was filled with wonderfully mathematically based knits.

I've written about this topic before but all the pieces were spread out all over the internet but these projects are all in the "Creative Knitting special issues" from April 2017 that is filled with products using 1, 2, or 3 skeins.  I had a blast reading it mathematically.

It started off with something alledan infinity cowl.  It might be because it offered multiple ways to wear the cowl so each time it was put on but that wasn't the one that caught my attention.  It was the Ridged Moebius cowl, one could knit that is large enough to wear over the shoulders.  If you didn't like that one, you could always make the Roman Stripe Moebius. 

A few pages later in the issue, I stumbled across three different sized Modern Cubist Baskets which are either proper cubes or a rectangular prism with handles that one can place knitting in.  The bottom and sides are knitted separately knitted before being attached.

Immediately following this project, I found instructions for making nautical coasters in circles, hexagons, and squares.  The authors even offer a square within a square.  Of course, if coasters are not the thing, then try potholders which are square in shape but have fascinating patterns such as a pot holder made of blocks created by the pattern.

Next come the instructions for making felted bags in a squarish shape.  They are first knitted, then felted with handles that gather the mouth of the bag closed. This is followed by headbands that are nothing more than long rectangular strips of knitted material attached to form a circle.

Talk about being in heaven, Moebius strips, cubes, rectangular prisms, squares, hexagons, circles, and more.  So many mathematical concepts rendered in knits.  Furthermore, knitting is filled with patterns, patterns, and more patterns created to take one long piece of yarn and turn it into something so much more.

While I am out at Christmas time, I think I might stop and pick up some needles and yard to make these things over the next few months.  No I'm not a great knitter, I'm able to follow patterns well enough to make a pair of socks but I'm not quite up to making lace. 

There are enough patterns in this book and on the internet to have for a after school mathematical knitting club.

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

Warm-up

Warm-up

School Based Business or Store.

  The business teacher at school is new but brings with her a plan.  She went to Donors choice to get the funding for equipment to cook hamburgers and fries so one of her classes could learn to run a business.

The idea is for all the students in that class to obtain their food handlers licenses to prepare them for jobs after school.

In addition, they have to figure out much meat, buns, fries, sliced cheese, and condiments to order for an event.  This means, they are using math when they fill out the purchase order because everything has to be ordered in from Anchorage and shipping can be quite expensive because it is air freighted.

Another thing they end up doing mathematically is calculating the amount they can sell the hamburgers and fries for so they make a profit and so people will buy their product.  Its a fine line sometimes because if the price is too high, they may not sell everything and loose money.

Out here a good hamburger can go for $25.00 with fries costing $5.00 for some but the school isn't charging that amount because they feel they can still make a decent profit without going that high.

Furthermore, we have a concession stand to sell things like soda, chips, and candy to  people at various activities, sports games, etc.  The senior class has it during the first semester while the juniors take over for the second semester.  Students usually sit down with the local Span Alaska catalogue to buy things by the case.

Span Alaska is a company who has been selling case lots and individual cans via catalogue in Alaska since 1972.  The price they quote usually includes the shipping already tacked on but if it requires special handling or is frozen, shipping is added on. 

Students order soda, gatorade, candy, chips, and other goodies to sell in the concession stand.  Since the price includes shipping, they know what the total cost is for the purchases so its easy to determine the cost they paid for each item.  Based on this, they can figure out how much of a mark-up for the selling price but they also have to keep track of how much the item sells at the local stores.  If they are too much out of line with the rest of town, many people will pop over to the store to buy things.

The idea is that students use this as a fund raising activity for their last year in preparation for graduation.  If the class has earned enough money over the four years, they might take a trip to Hawaii or to California.  Its a great experience because they have to determine how much it will cost to fly to a certain destination, rent cars, rent hotels, include spending money, etc.  In other words they have to create a budget and a goal.

Although both projects require a lot of work, it gives students a great experience in real life application of math, especially math geared for running a business. Let me know what you think, I'd love to hear.  Have a great weekend.

The 2018 Cost of the 12 Days of Christmas

I grew up listening to the 12 days of Christmas, both the traditional version and the Hawaiian version.  I'll admit that for the longest time, I thought it was 3 French Horns because I'd never heard of a French Hen.

It was only after I started to read, I discovered how far off I was.  Around Christmas time, it is possible to find the current cost of giving the 12 days worth of gifts.

This year, I managed to find a wonderful article in Forbes which included the increase or decrease for each item. According to the article, published in November, it will cost $39,094.93 to buy all these gifts.  This is an increase of $450 over last year which comes out to a 1.2% increase.

The list, including increases is:

•  1 Partridge in a Pear Tree: $220.13 (+.1%)
• 2 Turtledoves: $375.00 (no change)
• 3 French Hens: $181.50 (no change)
• 4 Calling Birds: $599.96 (no change)
• 5 Gold Rings: $750.00 (-9.1%)
• 6 Geese-a-Laying: $390.00 (+8.3%)
• 7 Swans-a-Swimming: $13,125.00 (no change)
• 8 Maids-a-Milking: $58.00 (no change)
• 9 Ladies Dancing: $7,552.84 (no change)
• 10 Lords-a-Leaping: $10,000 (+3.0%)
• 11 Pipers Piping: $2,808.40 (+3.5%)
• 12 Drummers Drumming: $3,038.10 (+3.5%)

Much of this information comes from the PNC Christmas Price Index which for the past 35 years has calculated total cost complete with increases and decreases.  The method used to do the calculations is almost the same process as the government uses to calculate the Consumer Price Index.

Their list is done for each item with a infographic like graphic for each item along with the percent change.  Furthermore at the bottom of the webpage is a graph showing the increase of the cost, year by year, beginning in 1984.  This is a great graphic to read and interpret.

In addition, at the bottom of the page is a 12 page activity guide with everything needed  to teach lessons on this topic.  The activity includes reading data, estimating, graphing, and requires students to explain their rationale for certain things.

I know what I'll be doing next week on the last couple of days before holidays.  Let me know what you think, I'd love to hear.  Have a great day.

Why Use Exit Tickets?

We see recommendations to use exit tickets regularly in class but one often wonders why use them in the math class.  Why even use them at all because its one more thing to use in class because its just one more thing to keep track of in class?

Exit tickets are like thermometers where the teacher checks overall student understanding.  They can be used to check for student understanding on a topic so the teacher knows if they need reteaching or if they are ready to move on.

In addition, the exit ticket provides information on if the whole concept needs reteaching or if a small point needs clarification such as if you multiply or divide by a negative, the sign changes.

Furthermore, exit tickets help students understand that the material is important and they are accountable for learning it.  It helps them synthesize the material, helps them move it from short term to long term memory because they are accessing it.

Another plus for using exit tickets is that students have to learn to communicate in writing.  If the teacher checks answers and has questions, they can clarify points to understand student thinking better.

If you do not currently use exit tickets, it is recommended you start slow.  Perhaps use it once a week and only on one or two topics.  It is suggested teachers do not grade exit tickets because its an assessment tool designed to provide data for instruction.  It lets the teacher focus on students who still don't quite have it with a bit extra instruction while provided more advanced problems for those who "have it".

To create an effective exit ticket does not take much.  Just follow a few simple rules:

1.  It is linked to the objective of the lesson.
2.  Focus on one skill or concept taught that day.
3.  Questions may be multiple choice, short answer, or require a couple of sentences.
4.  Exit tickets should have no more than 5 questions but fewer are usually better.
5.  Students should be able to finish the ticket in a few minutes.

The questions should not require a simple yes or no answer because that give no information.  Exit tickets should have questions that assess understanding, allow student to demonstrate the concept through work, or application of the concept.  You might create a problem based on the day's topic that could show up on a test redone, discuss how the topic could be used in real life, rate your understanding of the topic based on a 1 to 10 scale, or write a short paragraph on the day's lesson.

In addition, it is quite easy to set up digital exit tickets using google forms, or other app so you don't have tons of paper to keep track of.  More on this topic in a while.  Let me know what you think, I'd love to hear.  Have a great day.

Multiple Choice Questions - Other Uses.

When most of us hear the term "Multiple Choice" in connection to math, we automatically joke "Multiple Guess".  I remember taking a few where I just plugged the answers back into the problem to find the correct choice, or I used the elimination method to narrow the choices until I got it down to the most probable one.

Many multiple choice questions are set up so the most probable incorrect answers are included so if someone makes a mistake, they'll find it in the choices.  More often than not, you aren't sure if they know the material, got tired, or didn't care.

The other day, I read a blog on how to use multiple choice questions in ways that may be more effective in class. There are ways to use it other than as a major test.  I liked what the author said so I'll share it with you.

I'd like to thank Pear Deck for these suggestions  which are easily integrated into math.

1.  Use multiple choice questions to poll students on a topic or get feedback on something.  You might ask them if they'd prefer to play kahoot or jeopardy as a way of reviewing the material in preparation for a test.  You might also ask which step is next in a problem by listing several steps to  see if they understand the process. 

2.  Multiple choice can be used to check for misconceptions and understanding in topics such as
      GCF & LCM
      Equivalent Fractions
      Order of Operations
      Binomial Multiplication
      Combining like terms
      Solving problems
The use of whiteboards either virtual or real is great for this because you post the question, students write their answer on the board and with a quick glance, you see who understands it and who needs a bit more work.

3. Use multiple choice as an exit ticket.  Ask how they feel about the material studied that day using emoji's and multiple choice.  Faces can range from great to crying so students can choose one to show how they feel about their understanding.

These are just a few suggestions for using multiple choice to learn more about student understanding outside of a testing situation.  I think they are cool and add to my teaching toolbox.  Let me know what you think, I'd love to hear.  Have a great day.

Compare and Contrast in Math

Compare and contrast is something often used in Social Studies, English, or other subject but it is not something used as often in math, not because its too hard but because most math teachers have not been trained to use it.

The closest thing we have is the Vann Diagram but its not exactly a compare and contrast exercise.  So with a bit of thinking and working, it can be used in the math class but not for every topic.

First its important to know that when compare and contrast is used, it can strengthen the student's memory, help develop higher order thinking skills, improve comprehension, precision and helps build good work habits.

Lets look at some ways to use compare and contrast in math.

1.  Compare and contrast inequality signs.  Many students have difficulty in distinguishing between less than, more than, greater than or equal and less than or equal.  Requiring them to fill out a chart comparing similarities and contrasting or finding differences can help them put into words their understanding.

2.  Compare and contrast place values such as tenths and tens, thousands and thousandths, so students learn to tell the difference between decimal values and whole number values.  They are very similar but with a small difference.  Some of my students arrive in 9th grade not knowing their place values which hurts them at times.

3. Long division and synthetic division in Algebra.  Although they appear different, the methods share quite a few similarities.  Using a compare and contrast will show students their similarities and differences.  In addition, they can see that it doesn't matter which method used, the answer is the same.

4. Compare and contrast distributive property with binomial multiplication.  It doesn't really matter what method you use to complete binomial multiplication but it still involves distributing terms, just like we teach when using distributive property.

5.  Compare and contrast Greatest Common Factor with Lowest Common Multiple. I can tell you, my students are having so many problems with this topic so completing a compare and contrast in the hopes they can learn to distinguish between the two.

6.  Compare and Contrasting congruent with similar triangles since both share similar methods of proof. 

7.  Compare and contrast bisectors with medians and altitudes because students often have trouble remembering the differences among them.

8. Compare and contrast area with surface area.  I realize one is 2 dimensional while the other is 3 dimensional but they do share similarities.

Any two math topics which have similarities and differences can be used in a compare and contrast exercise.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Warm-up

December 7, 1941

Today is the day to remember an event that happened 72 years ago.  Japan bombed Pearl Harbor, killing both military and civilians while trying to cripple the American fleet.

Its sometimes difficult to find activities designed specifically for the math classroom.  Most of the ones I've found are for history or English but that has not stopped me at all.

This particular day has lots of possible activities which you can provide the data for, or you can have them find the data for the project.

1.  Its easy to find the number of people killed in each branch and civilians on December 7, 1941.  This information can be turned into a chart to show the information visually or it could be turned into an infographic with the graph.

2.  Another graph could be done showing those from each branch and civilians who were wounded during the attack.  You can find the information from the Honolulu paper showing how many and the ages of civilians who were killed by bombs or bullets from the attack.  Again, its easy to create a graph showing age distribution.

3.  Number and types of ships that were damaged or sunk. The Japanese wanted to sink aircraft carriers but there were none in port that day.  Instead they got battleships, destroyers, etc. This exercise could also include the number of planes destroyed or it could be done seperately.

4.  Find out where the mini-subs departed from the mother subs, when they left and when they arrived at pearl harbor.  This is enough information to calculate their average speed or you might find the average speed to determine how long it took them to travel the distance.  Furthermore, its possible to calculate the volume of the two man mini-subs.

5.  Another activity dealing with distance is to find out where the Japanese planes took off from, find the distance, and how long they took to get to Pearl Harbor to determine the average rate of speed. 

6.  Find out how many total planes the Japanese sent to bomb Pearl Harbor and the number of each type of plane to create a graph.

7.  Students can also create an infographic containing all sorts of numerical information on the attack on Pearl Harbor. 

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

Modeling Viral Fake News.

It used to be, people would open a newspaper, watch television, or listen to the radio to discover what was happening in the world.  If you were lucky, you might see it happen live on television such as during an earthquake when the world started shaking and you saw the newscasters dive under their tables for safety.

Now, news is more instantaneous with the use of social media.  You can see things as they happen because people can record and instantly post on any one of the numerous social media sites.  You can also see people add 2 + 2 to get 5 instead of 4 and this is when we see fake news going viral.  There are people out there who have worked out the math on why fake news goes viral.

First of all, the way social media is set up, just about anything can go viral due to the amount of information out there and the inability of people to fully evaluate every piece they see.  Most of these pieces of "news" include a video, picture, link, phrase, or other form of online information and the "reader" has to sort through so much.  Second, there is the amount of time people spend looking at any item so if it doesn't capture their attention, they'll skip it and share something that does.  Finally, the way social media is set up, it encourages  indiscriminate sharing.

Now as far as the math goes, social media uses agent-based models because individuals are the ones who share things.  This is the same model used to determine how disease spreads through communities.  If you were to visualize it, you'd see dots representing the individual and arrows pointing from the individual to others as they shared the disease or fake news.

Mathematicians have had to modify it a bit because the original model is for one disease not thousands of pieces of information shooting across the internet each day so they've had to include the probability of a person making a new piece of information or looking through things they've gotten before sending it on.  In other words, they look at the most likely attention span of the individual for sorting through all the messages before sending one on.

People have speculated that super connected people on social media are more likely to cause something to go viral but one scientist looked at that speculation and concluded it is not true.  She stated, most of these super connected people do not have time to go through all the material they receive and they certainly don't have time to send everything they might want to.

What is more likely is that groups or clusters of people are more likely to socially share, creating a social reinforcement of material because every time you see it, it becomes more believable.  If its believable, it has to be true right?  This is one reason certain pieces of fake news go viral.

Scientists and mathematicians are still working out the intricacies of this topic but they are getting closer.  Let me know what you think, I'd love to hear.  Have a great day.

Math and The Stock Market.

Now that you know why the stock market used fractions, its time to look at the math involved in the stock market and there is quite a bit.

First off, anyone who has a portfolio with a broker or of their own, knows the portfolio is divided into a pie chart showing how much is in stocks, bonds, cash, and mutual funds.  If your portfolio does not have that much diversity, it could show how many shares of each stock you've invested in.

Second, it is possible to calculate the return or money you are making off your portfolio by using this equation.  (Current Value/Starting Value - 1) * 100 tell you how much you made or lost for the entire portfolio.  This is important because so many people are interesting in building a portfolio which will support them for their lives, throughout their retirement.

Third, people calculate the stock return so they know if a stock is performing poorly or well so they know if they want to keep it or sell it off.  The formula is similar to the last one but this time its
(Last price quoted/price paid - 1) * 100.  This formula can be applied to each stock to determine its return so you know if you want to keep it.

Here are other equations associated with the stock market that investors need to know.

1. Earnings per share is a part of a company's profit set aside for common shares which is an indicator of how profitable a company is.  The formula (Net Income - Preferred Dividends)/Total Outstanding Shares.

2. Return on Equity also known as a rate of return on net assets often is associated with the company showing its financial performance.  It is found by (Net Profit/Shareholders Equity * 100 ) which give a percentage. 

Now if you check this site out  there are lessons on Risk and Return, Return on Investment, Investing Options, Stocks and Stock Market, Stock Investment Analysis, and Bonds.  The Stock Market lessons are quite interesting because there are lessons on investing in stocks, stock market table, stock market simulation, buying shares and determining how many shares one can afford, percent price change in shares, and forms to fill out for buying and selling stocks.

In the Stock Investment Analysis, there are exercises for price earnings ratios for stocks, and common stock valuations while the Bonds section has information on buying and investing in corporate bonds.  Information that few people really understand.  Even I have someone who does all that for me because I don't know how to go about it.

The stock market simulation uses most of the information provided to give students an experience in investing in the stock market.  This activity could take as little as a week but its recommended the simulation be run the full year to give students a better understanding of how it works.  Throughout the designated time period, all students invest a specific amount in 5 stocks which they will follow over a certain time period.  they may buy and sell but they must pay a 3% commission each time they buy or sell a stock.

They have to fill out transaction forms every time they buy or sell stock, monitor the portfolio and if you want at this point, you could have them calculate the overall worth increase or decrease of its value each week.  This process continues until the end of the period at which point they sell off all the stocks to determine how much money they made or lost overall.

Its a nice basic simulation that you as teacher could add to so students are doing a bit more mathematics.  Its up to you.

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

The Stock Market Uses Decimals Too!

  When I was in school, one of the common examples of needing to know fractions included the stock market because they would show everything in eights.  For instance Apple might go up 1 3/8 points while General Electric might drop 1/8 of a point.

Just a bit of background on the stock market and why it used fractions rather than decimals originally.  The New York Stock Exchange was formed back in 1792 due to the Buttonwood Agreement.

 24 leading bankers, brokers, and merchants agreed to create a central clearing house for trading stocks and securities.  These men modeled their exchange on the one in Spain after checking out others in Europe because the United States currency had been based on the Spanish Real.

The Spanish silver dollar or Real was divided into two, four, or eight parts which is where the term "pieces of eight" came from because they could count them on their fingers. The Spaniards did not count using their thumbs like the English did.  So when the stock market began, they based the smallest increase on 1/8 of a dollar or 12.5 cents. In other words, if one stock dropped 1/8th you'd only loose 12.5 cents but what if you had 100,000 shares of the stock you'd loose $0.125 x 100,000 or $12,500 which adds up. 

This spread could cause people to gain lots of money when dealing in millions so they dropped the increase or spread to 1/16 or 6.25 cents.  Along the way, some stocks used a spread of 1/32 or 1/64 to keep it much smaller.

In 1997, the Common Cents Stock Pricing Act was passed by congress to make it easier for people to understand the pricing system since more and more people were investing in it.  Although, stocks began changing over in August 2000, it took till February 2001 for all 3025 companies listed on the NYSE to convert from fractions to decimals.  This change encompassed about 280 million shares.

Two of the things this did was to:
1. Investors could save over $1billion or more each year.
2. Investors could save on the cost of commission since commission was often based on the price of the stock.
3.  The United States is more compatible with other stock markets who have been using decimals for years.
4.  The number of transactions handled by the NYSE is able to double using decimals instead of fractions.

This was just a look to see why the stock market used fractions for most of its life and why it switched to decimals.

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

Moving From LCM to Adding Fractions Using Legos.

 As you know, I've been playing with Legos to figure out how to use them for something other than basic fractions in lower elementary.

I already figured out how to use them to help find Lowest Common Multiples or LCM so its really not a huge step to seeing why its important to have equivalent fractions when adding.

So I used the same number of Legos for the denominator and used a second set of Legos in a different color to represent the numerator. 

As you can see in the photo above, you see the 1/4 and 1/6 quite easily.  Then using the same method I added another 1/4 and 1/4 to get 3/12 which is the correct equivalent fraction.

For 1/6, it required a second 1/6 to get a total of 2/12.  This allows students to easily see why you multiply the numerator and denominator by the same number so you end up with equivalent fractions.

The same process could be used to show subtraction as well as addition so students can see it.

If you have enough Legos, it is possible to have larger denominators such as 15, 28, or 34 and allow them to see the process works as well on larger denominators as for smaller ones.

Let me know what you think.  I'd love to hear.  My next project is multiplication of fractions and whole numbers using Legos.  As soon as I have that done, I'll share it.

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Warm-up

8 Ways Probability is Used in Real Life

Probability is one of those topics we teach in a rather isolated manner.  I know for me, its thrown in here and there as I can fit it in but I never really address where we see it in real life.  As a matter of fact, this is another topic that would make a good infographic.

Lets look at some of the places students experience it in real life and might not even be aware of it.

1.  Weather reports usually give people the probability of rain, snow, clear, or fog.  I've listened to them but never thought anything about it.  If you hear that there is a 60 percent chance of rain, what that means is that same weather conditions produced rain for 60 out of 100 days.

2.  In sports, coaches look at certain statistics to determine who to play in a specific situation.  For instance if a baseball coach has two possible hitters, one has a .200 hitting average or 2 hits every 10 times at bat versus a player with a .400 hitting average or 4 hits every 10 times at bat.  Who would he choose?

3.  Insurance uses probability to decide how much to charge and people also look at the frequency of something happening to decide if they should get extra coverage.  One example is when you decide to get liability or comprehensive and liability based on how frequently something happens.  If 18 cars out of every 100 hit a deer, you might want both because there is an 18 percent chance of hitting one but if there is no chance of hitting a deer, you might only go with liability.

4. In games, people look at the possibility of getting something such as in poker there is about a two percent chance of getting three of a kind or a 42 percent chance of getting one pair in a hand of poker.

5.  Advertisers use probability to determine who should get which type of coupons such as looking at a woman within a certain age group is more likely to have a baby so send her discount coupons for diapers.  This is called targeted advertising.

6.  Horse racing tracks use probability to weight the betting odds so the race track owners cannot loose.

7.  Machine learning uses probability to determine the chances of the next word is as you type a text on your phone or if you type a word in a search engine, the search engine uses probability to determine the one you want.  For instance if you type in the word "cricket" it will show the most probable meaning such as the insect, the game, or the phone company.

8.  Stores use probability to order and stock shelves so they don't run out.  For instance, they want to stock up on turkeys, ham, cranberry sauce, and other traditional goods for the Thanksgiving/Christmas season or chocolates and eggs for Easter.

There are so many more ways probability is used in real life but these eight are just a quick look at how probability is used in ways we don't think about.  Have a great day and let me know what you think.

Using Legos to Find LCM

A  4 block and a 6 block.

 As you know, I've been working on ways to use Legos in the high school math classes.  Sometimes, we have to teach skills to students who are supposed to have learned them prior to arriving in high school but didn't for whatever reason.

I've looked at scores for my students and too many are testing in at a 4th, possibly 5th grade level and have managed to get passed grade to grade without the skills.

My pre-algebra class is on their second week of learning LCM because I discovered that too many have been confusing GCF and LCM.

I've tried the fraction strips, listing multiples, factoring trees and many of my students are struggling to learn how to find the LCM.  Some one in elementary taught them something called the butterfly method where they cross multiply the numerator and denominator of two fractions and then multiply the denominators together for the common denominator so they multiply the two numbers together to get the LCM.

My objection to the butterfly method is that it only works if you use two numbers which are some of the smaller numbers or contain a prime such as 4 and 7, otherwise if you have 8 and 4, you end up with a LCM that is too high.  Yes I'm mentioning fractions because LCM and fractions go hand in hand.

I've added Legos to find LCM.

So I pulled out my small set of Legos and started playing with them.  I chose 4 and 6 because those are some of the multiples students have to work with when first learning the process.

First thing I did was lay out a block with 4 circles (a 2 by 2 block) and for 6, I used a four block and a two block to make 6.  You can see it in the first photo.

Second, I added enough four blocks and six blocks until they were exactly the same size as seen in the photo to my left.

They can see they need three 4 blocks and two 6 blocks to get to the lowest common multiple. I think this may provide a better visual than the strips or the butterfly method.

I am hoping they "see" how the numbers relate.  Only time will tell.  Let me know what you think, I'd love to hear.

Ways to Use Infographics in Class

Yesterday I went over the elements needed in a good infographic and today its time to look at how they can be used in the classroom.  Since the number of infographics has increased, its important students learn to read and interpret the information contained in them.

Its also important students learn to use infographics in as many ways as possible.  So lets look at ways they can be used.

1.  Find infographics which compares information to inspire discussion and debate.  The inforgraphic should show two sides of the question such as wild vs farmed salmon.  

2. Let students create their own infographics on a topic such as Black Friday sales or Flowers sold on Mother's day.  Creating infographics helps improve both computer skills, creativity, and critical thinking skills.

3.  Use an interactive infographic as a quiz or activity in the class so students have to work their way through it.  This type of activity improves literacy, math skills, and reading ability.  This is one way to gamify the classroom.

4. Use the infographics as visual aids in the classroom because they combine images with useful data so its short and sweet.  The infographics contain only essential information in an organized form and meets several standards on learning to read information presented in a variety of forms.

5. Rather than having students write down what they know on a topic, have them create an infographic instead because they cannot fall into the trap of writing everything they know in the hopes of getting some points.  They have to get to the essence of the material.

6. Instead of a homework assignment, assign them to create an infographic.  It might be on the three ways to solve systems of equations and the process to select which one is best in what situation.

7. Infographics can be used as the visual for a presentation so all the information is in one place on one page, thus eliminating that awful moment when nothing works or it freezes.  The infographics allow the presenter to have a flow to the presentation based on how they put it together.

8. Use the infographic as a way for students to report on a topic.  It has been found that infographics can be used to transfer information about a topic faster and more effectively than straight text as long as the inforgraphic is well designed.

Yes, these can all be used in the math classroom with just a bit of thought.  Let me know what you think, I'd love to hear.  Have a great day and I do plan to revisit this topic later but tomorrow, LCM's and Legos.

Infographics?

Yesterday and this past Friday, I suggested students create infographics on the sales for Black Friday and Cyber Monday. Aside from the fact that infographics are becoming more and more popular, one may wonder why we should teach this skill in math classes.

By definition, an infographic is a quick way of presenting information, graphs, or data in a visual way that makes it easy for the brain to see patterns. 

Furthermore, a well designed inforgraphic makes it easier to communicate data to others.  In addition, infographics can be used to introduce new topics, spark discussions, or provide a starting point for more in depth research.

In a sense, creating an infographic is like writing a paper.  The author has specific information they wish to convey to others.  They need to choose the right visualizations or images,  good color combinations with proper spacing and boarders to attract the eyes and make it easy to read.

There are three main parts of the infographic.  First is the visual made up of color coding, graphics and reference icons.  The second is the content with time frame references, statistics, and information on where the statistics were found.  The last part, is the knowledge with facts and deductions.

One way to help students learn to create good infographics is to find some and share them with the students one at a time.   Ask the students to analyze what is good about each one, what could be improved.  Let them discuss what each designer did well in regard to creating the right balance between graphics and text.  Ask if these infographics have all three parts and if so is the information in depth or simple.

Once they have had a chance to see a variety of infographics, it is time for them to think about creating their own. There are five steps to think about when they begin designing their own infographic.
1.  Create a flow chart or skeleton of the information so students know how the information should be grouped and how it is related to each other.
2.  Assign a color scheme so the information is easy to see and is not overwhelmed by a mish mash of color.
3.  Choose the appropriate graphics that tie everything together.  If you need icons, this is where you think about them.
4. Research the data to make sure it is correct and supported by sources.  The best ratio for data to graphics is 1:1 so you want to make sure you have an equal amount of data and graphics. Furthermore, make sure you know who your audience is because that will impact much of the data and graphics you choose to use.
5. Make sure the knowledge is placed so people will be able to make deductions easily.  Do not make someone feel stupid when they read it.

 It is important to plan this all out on paper prior to creating the actual infographic much like a film maker uses a story board to plan a film or animators use a story board when animating a cartoon.  Its best to plan it all out in detail.  Keep your eyes peeled for ways to use infographics in class as part of your teaching.  Today was focused on helping students learn to create their own but they can be used as part of your teaching.

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

Cyber Monday Stats.

The term Cyber Monday has only been around since 2005 when it was coined to describe the buying done on the Monday after Black Friday.

This date provides a great set of information for students to make another infographic or perhaps a wonderful graph where students can calculate the yearly growth.

 I obtained the information from this website.

Cyber Monday spending for the years 2005 to 2017.

2005: $484 million
2006: $608 million
2007: $733 million
2008: $846 million
2009: $887 million
2010: $1.028 billion
2011: $1.25 billion
2012: $1.46 billion
2013: $1.74 billion
2014: $2.04 billion
2015: $2.28 billion
2016: $2.67 billion
2017: $3.36 billion

In 2017, the purchases on Cyber Monday were as follows:

22% sought deals on clothing.
21% wanted deals on tablets/laptops/PCs/TVs
17% looked for deals on smart-home gadgets.
15% wanted deals on gift cards
14% preferred looking for deals on toys.
11% sought deals on travel.

The top two retailers who profited from Cyber Monday are:
Amazon who secured about 60% of all Cyber Monday sales via 108 million visitors and 8 million transactions.
Wal-mart was second with only had 8.5% of the sales via 32 visitors who made 1.2 million transmissions.

A few facts. 
1. In 2017 81 million people shopped online on Cyber Monday.
2. Cyber Monday has become more popular than Black Friday with a 71% to 69%
3. 75% of the shoppers used a home computer, 43% used a mobile device and 13% used computers at work.
4. 88% in the 18 to 34 age group planned to shop on Cyber Monday while 74% in the 35 to 54 age group planned the same.

Have fun letting students loose with this information to create graphs and infographics so they can learn to communicate mathematical information.  Let me know what you think, I'd love to hear.  Have a great day.

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Black Friday Project.

I just realized how perfect Black Friday is for a project in Mathematics.  One of the dictates of math is for students to learn to communicate in a variety of methods.

One such method is to use infographics to communicate information in a visually satisfying manner.

In addition, infographics often use statistics as the basis for information included.  One time, I had fun creating one on what happens the first day after going onto Daylight Savings time.

Black Friday, besides being one of those sales days where stores open at 5 or 6 in the morning, is full of statistics perfect to communicate via infographics.  Look at these numbers:
1. In general 30 percent of yearly retail sales occur between Thanksgiving and Christmas.  In some areas such as jewelry, it can be 40 percent of all sales.

2. In 2015 - 74 million people shopped in stores on Black Friday
    In 2016 - 101.7 million shopped in stores on Black Friday
    In 2017 - the figure declined 4 percent.

3. In 2015 - 102 million shopped in stores for the whole 4 day weekend.
    In 2016 - It jumped to 134 million.

4.  In 2017 - 7.9 Billion online sales.  40% were made from mobile phones
     In 2016 - on line sales were 18% less.  calls from mobile phones were 29% less.

5. On average each shopper is expected to spend an average of $1007.24 which breaks down as follows:  $637.67 on gifts, $215.04 for food, decorations, etc, and 154.53 on seasonal deals.

6. Based on 15 years of data, annual sales increased on average 2.5 percent.

7. Data found here for 15 years worth of sales beginning in 2002. This information could also be used to create graphs showing the overall increases and decreases for use in the infographic.

8.  In 2017 between 500,000 and 550,000 seasonal workers were hired.
     In 2008 - 263, 820 workers were hired.
     In 2013 - 764,750 workers were hired.

Lots of great data to convert into infographics.  Let me know what you think, I'd love to hear.  Have a great weekend.

Happy Thanksgiving

Farmland as an Investment.

  I was reading my favorite magazine the other day when I came across an interesting article on how farmland is considered a great investment based on its return. 

In 1970, an acre of farmland in Iowa ran just over $400 but that same acre jumped to just over $7000 in 2016.  that is a huge increase.

(7000-400)/400 = 1650% increase per the standard mathematical increase formula.

According to the article, farmland as investment has outperformed most other assets for the last two decades offering a 12 percent return each year.  This has lead to investors bidding on proven properties for their portfolios.  So far about 30% of the available land is owned by investors and leased out to farmers.  

You may wonder how this happened.  Its simply economics because many farmers are aging, The cost they can sell crops is less than they need to be financially solvent, the interest rates have gone up, and people want their money out.  In addition, some crops such as corn are being used as an additive to fuel.  Interesting observation, as the price of land has escalated, the amount of ethanol being produced has grown at about the same rate.

Over the next 5 years, according to another article it is predicted that another 92,000,000 acres of land will come up for sale, or an amount of land that is about the size of Montana.  It is expected this land will be snatched up by investors such as retirement or pension plans, and others rather than being purchased by individuals.

In 2016, the total amount of land in the United States under crops is 253.1 million acres.  This means if 30 percent is owned by investors, then 253.1 x .30  or about 76 million acres so that would be (92 + 76)/253.1 or about 66% of the farm land owned by investors.  On the other hand, I saw a different figure which stated only 10 million more acres would be available over the next 5 years for purchase so that would be (76 + 10 )/253.1 or about 34 percent. 

Both articles agreed on the amount owned by investors but differed on how much will be available for purchase over the next five years, I looked at the percentage for both.  What I see is that this article is great for having students calculate the actual numbers of acres owned by private investors so they could take this information and create a slide show using the information.

This helps teach them more about the rate of return, economics and provides students with a real world application of mathematics.  Let me know what you think, I'd love to hear.  Have a great day.

Scale models.

  Usually in most textbooks I've used, problems involving scale models all seem to revolve around planning a garden, drawing plans for a house, or even figuring out the length of a real car or train based on a scale model but there are other things we can use which might provide students with an understanding of other subjects.

For instance in science, they learn about the layers of the atmosphere, peer at a diagram in the book and move on but what if we took time to help them draw a picture in math while studying it in their science class?  Might that show students there is a connection between the two classes?

Since most of the world uses metric including the science class, its best to draw this on centimeter grid paper so students become used to using metric.   First step is to have students draw a circle with a radius of 6.378 cm in the center of the page.  This represents a scale of 1cm = 1000km so a radius of 6.378cm represents a real radius of 6378 km.

99.999% of the atmosphere is within the first 100 km above the planet so you have students draw a small line about 1 mm above the earth.  That shows the students a reasonable visual of how thick the atmosphere is around the earth.

More specifically if air pressure is 1 kg per square cm at sea level.  If you move upwards to around 5,500 meters or 5.5 km, the pressure drops to half of what it was at sea level.  Go up another 5,500 meters or 5.5 km, its dropped in half to about one fourth of what it was at sea level.  Most planes fly between 11,000 and 13,000 km and if you get up to 30 Km above the earth, you are above 99 percent of the atmosphere.

If you have the students create a thin strip to show the layers of the atmosphere, the tape from an adding machine is great for that purpose or a strip of graph paper with 1 cm = 1 km works equally well.  The layers would be as follows:

1. The troposphere is about 15 km thick.
2. The tropopause is about 5 km thick.
3. The stratosphere is about 30 km thick.
4. The mesosphere is around 40 km thick.
5. The ionosphere is 260 km thick.

This activity shows how the atmosphere is divided up but its nice for students to see that the whole atmosphere is not that thick. 

Each activity involves creating scale models, each with a different scale.  It would be quite difficult to see the different layers of the atmosphere if the scales were the same.  I believe this shows the need for different scales.

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

More on Making Thinking Visible.

  Many students find it difficult to explain their thinking when it comes to mathematics.  Often they can tell you what the next step almost by rote but they cannot always tell you why they need to do it.  This can be especially difficult for ELL students who may not be as fluent as other students.

Some students when you ask to explain their thinking often give you a look like "Why do I need to think about it?  I just need to follow the steps and I'll get a right answer!"

They don't seem to understand that thinking leads to better understanding.  There are ways to encourage and help students develop the ability to explain their understanding.

1.  Connect-Extend-Challenge which adds a layer to instructing students in a new concept or skill.  Instead of just teaching the skill, ask them how it connects to other problems they have solved because it requires them to access their prior knowledge.  Once they begin connecting the problem with other problems, ask them what is new before asking them to explain how this problem extends their knowledge and thinking.  The final step is to have them identify the challenges they faced learning the material.

2. Claim-Support-Question is designed to help students make a claim, recognize patterns, figure out generalizations, and learn to provide evidence to support the claim.  The teacher makes a claim such as "all multiples of four are also multiples of two".  Students gather in small groups to discuss if the claim is true or false.  They must provide evidence such as manipulatives, drawings, etc to support their position on the claim.  The last step is to have students list the questions that were raised during the discussion that have not been answered.

3.  See-Think-Wonder is for students to look at a visual pattern while explaining what they see.  They talk about it and then think about the next step in the pattern before discussing their choices.  The final step would be for student to express what they wonder about in regard to the pattern.  Do they wonder what the 100th step is or perhaps the 50th?  This is where they talk about it.

These are just three routines which can easily be implemented into your daily routine to help students learn to make their thinking visible and to help them increase their conceptual understanding of mathematics.

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

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