Warm-up

Book Creator

  Last night,  I had a blast participating in a one hour web instruction where I learned to use the Chrome based version of Book Creator.

This app/web based app allows one to create a book with photos, comic strips, videos, text, and other things.

After a short hour with 5 tasks, I feel comfortable enough to use it and help my students learn to use it.  It is easy and quite intuitive

 I am always looking for ways to integrate more writing into my classroom so my students have a chance to use it outside of English.

In addition, when they have to explain the math through writing, movies, and pictures, it helps clarify their actual knowledge and helps them practice their mathematical language.  In regard to the writing segment, they must use their language properly, be able to explain the concepts clearly, and provide proper examples.

As for visual, they need to create drawings or pictures able to convey the ideas clearly.  It is important for students to be able to do this to show they understand the concept behind the written representation.  Furthermore, if needed they can also create a video showing real world applications or creating a news report sharing the mathematical concept.

This way they see it, say it, and hear it as they create the book.  As the teacher you have a choice on how it can be done.  The students can each create a book or each student can create one to two pages for a book.  I like the option.

There is a way for the teacher to monitor student progress as they create their pages or books.  The price for 40 books in one library is perfect if your school is on a budget - free otherwise there are two options which allow more libraries and books but do cost something.

I am all for using apps which allows students to express their understanding of mathematics in more interesting ways than just solving equation after equation using processes without understanding the underlying concepts.

Check it out, have fun and let me know what you think.  Have a good weekend.

Brain Breaks

Again, I'd like to thank Twitter for this topic.  Its  new to me that I'd like to share.  It's referred to as "Brain Breaks" or taking a physical break to stimulate the brain into learning.

Brain breaks incorporate some sort of physical activity into the classroom for a minute or two every period.  A brain break is designed to clear the brain pathways of stress and overload so the brain is able to learn at its optimum.

It has been discovered that for the brain to absorb new material, the information must pass through an emotional filter called the amygdala before reaching the prefrontal cortex.

When students become to stressed out or overloaded with information, the emotional filter is activated to the point, information is unable to reach the prefrontal cortex.  Brain breaks are designed to help return the brain to a state of optimal learning.  Switching mental activity allows the brain to shift communications to fresh networks of neurotransmitters which allows the brain's chemicals to replenish within the resting networks.

As a general rule of thumb, brain breaks should occur after 20 to 30 minutes of concentrated activity for middle school and high school students.  The good news is that brain breaks need not interfere with instruction. A brain break may be having students do a quick stretch, have them move to another part of the room, or even put a short piece of music on to dance to.

I usually include some sort of movement in my classroom but not as regularly as I should.  Some of the things I do is to prepare several multiple choice questions for my Smartboard.  I have the corners of my room designated as A, B, C, D and they must go to the corner of the letter they believe is the correct answer.

I do not have students pass out work, I place any sheets, books etc away from the students and they must get up to get the assignment.  Yes, it takes a couple minutes but the students are up and moving.  In addition, I allow them to check notes for their tests but their notes are spread around the room.  If they want to check their notes, they have to get up, walk over, look at the notes before returning to their seats to work on the tests.  All tests remain on the desk.

I remember reading about a teacher who kept a ball in the room.  She'd have students stand up, ask a question, then toss the ball to a student to answer the question.  She'd ask another question and that student tossed the ball to another one to answer the question and so on.  The ball indicated the person had the floor.

Here are a few other suggestions I found that I like.

1.  Desk switch - have students grab their materials and find another desk to sit in for the rest of the class period.  By changing desks, they get a new perspective on things.  Be sure to set a time limit or it might take the rest of the period.

2. Sky writing - have students write a message in the air to a friend.

3. Give students a chance to play "Rock, Paper, Scissors" for two minutes while away from their desks.

4. Thumb wrestling.

5. Mirror - Mirror where students are broken up into pairs.  One student mirrors the actions of another student.

6. Pantomime - choose a student to act out an activity without talking.  Students have to mimic the leader and then guess what the activity is.

Let me know what you think.  I hope you all have a great day.

Real World Piecewise Functions

Up until recently, I never wondered when one uses piecewise funcitons in real life.  It wasn't important to me because I was only familiar with the mathematics.

After creating the hands on activity, it hit me, I didn't know when the piecewise function was used in real life.  There has to be some application because it exists since mathematics often explains the world.

So after a bit of research, I discovered several real world situations that are not contrived and make sense to me.

I assume most people reading this column have at some point worked a job where you were paid by the hour.  I had one where I worked filling the newspapers with sales flyers so I'd have days free to substitute and look for a teaching job.  The place I worked set the pay with the following parameters:  I was paid a per hour rate for the first 8 hours worked in a 24 hour period from midnight to midnight.  Anything over the 8 hours and I received time and a half for those hours.  If I worked holidays such as Christmas, I received double time.  That makes my pay a piecewise function because the amount I received depending on the number of hours I worked.

For many other jobs the piecewise is based on working a 40 hour week before a person begins to receive time and a half for overtime or double time for certain holidays or Sunday.  It all depends on the rules of the business.

Another piecewise function is based on the amount of something purchased.  The more of the one item you purchase, the less per item you pay.  I've dealt with a company where I paid full price for the first 10 items, the next 20 granted a 10 percent discount, etc.  I've seen this type of arrangement with t-shirts, water bottles, etc. 

Look at the launching of a rocket and follow its acceleration from launch to touchdown to see a piece wise function.  For the first two seconds, the rocket accelerates from zero to a certain speed.  Between two and twelve seconds, the rocket slows down as it begins to coast and due to gravity.  After 12 seconds, the parachute is released and the rocket floats back to earth. Based on the above situation, I think anyone who jumps out of a plane is going one velocity until they pull the ripcord causing the parachute to open and they are now floating down.

The next time I teach this topic, I can provide examples without students having to ask.  I feel like I've scored a win. 

Let me know what you think.  I love to hear from people.  Thanks for reading.

Mathematical Selfies.

  The idea for this column came from an article in the September 2017 issue of the Mathematics Teacher on mathematical selfies.

The idea behind mathematical selfies is to have students go out into the world to find examples of various mathematical concepts they can share.

For instance, pictures of a city skyline resemble a histogram, or a row of trash cans provided by the city show congruence since they are all the same.

 Take a picture of a mountain range and you have a wonderful example of a polynomial function.  Or a picture of a musical score can be used to show fractions.  Take a picture of shadows created by a house, apply parallel lines and transverals and you can identify a variety of angles.

The author, Kathy M.C. Juqua, teaches an informal Geometry and Mathematics concepts class.  Students chose to use selfies in a variety of ways. Over the semester, students are given 45 terms to work with.  Some found pictures representing different vocabulary terms to create individual visual dictionaries.


Others, alphabetized the terms and created a more traditional dictionary with definitions and pictures.  Some took the terms, divided them into groups so they could make connections between terms with their selfies.  In addition, students made study aids, and compilations.

Another teacher, Rachael Miller assigned her students to take  photos from their daily life, annotate them explaining a mathematical concept they are learning in math class.  The annotation provides a way to assess a student's understanding of the material.

She used it as a homework assignment.  At the beginning only a few students submitted their answer but after sharing the photos with the class, others began to turn their entries in until everyone was doing it.

I am glad I ran across both articles.  This might be a way to get my students more interested in mathematics by following Ms Miller's idea for a homework assignment.

Let me know what you think.  I'm thrilled to have read both of these works.  Have a great day.

PIecewise Function

The idea for this column came from a question posted on Twitter about ways to help students understand piecewise functions better.  I can understand the teacher's request for help because our students have difficulty combining two or more functions into one graph.

I gave it some thought and came up with an idea I thought I'd share with everyone.  It is a physical way to see how they combine.

First draw the two individual graphs on graph paper.  I made the one shown to the left.  I included the two functions, I listed as an f(x) and g(x).

I used two different colors so the graphs are distinguishable as possible.

In the second photo, I show the individual graphs against a white background.

This prepares the student for the next step.  Since the x^2 graph is used till the value of x = 1, the students will cut the graph at x = 1.

The second graph is also cut at x = 1 because that is where the second piece begins.

For the final step, have the students tape the two graphs together at x = 1 so they match up.

With the two colors, it is easy to see where one graph ends and the other begins.

At this point, students can put in the circle indicating the < part to see how well they fit.

The last piece would be having students fill out a small questionnaire in which they use the completed graphs to determine which graph is associated with which part of the graph.
I might ask "f(3) is found on which graph?" so they have to relate values to points and lines.

Students can repeat this exercise with several different piecewise functions to see how the functions fit based on the criteria.  Some will have a smooth transition like this one while others have huge jumps between one function and the next.

I also found this game called Polygraph: piecewise functions on Desmos.  It is designed for students to improve their vocabulary regarding piecewise functions, first by playing against the computer, then against each other only after answering questions designed to help them reflect on what they are learning.

I think part of the problem may be that students are not exposed to these types of functions until they hit high school.  I don't think its covered in middle school.  I do teach it but usually not until Algebra II because I'm usually bringing my Algebra I students up to where they should be when they enter the class.

In a couple days I will discuss the use of piecewise functions in real life.  When I took math in high school and college, the teachers never discussed the situations one runs into where they might have to use these types of functions.

Let me know what you think.  In the meantime, I'm working on ideas for sketch notes and graphing activities for linear inequalities and systems of linear inequalities.  I'll share those when I get them developed.

Thanks for reading.

Warmup

Warmup

Explaining Why in Math

  Recently, I've been requiring students to write down explanations of what they did or why they did something.  My students fight me on this but I'm noticing it is making them stop and think.  It shows me if they really understand what they are doing.

In geometry, my students are classifying triangles by angles and sides. In addition to stating its a right scalene, they have to add that it as a 90 degree angle with three sides of different lengths. 

Since my Pre-algebra class is still struggling reading signs during addition and subtraction, I started having them write down if there are two negative signs, a double negative, or different signs before doing the addition or subtraction.  They need to slow down and read it carefully.

In Algebra II, we are just starting finding solutions to systems of equations using the elimination method.  They are going to have to explain what they are going to multiply by to change the coefficients before actually doing it. They will also have to write down why they chose that. 

In another class I have students working on solving one step equations.  I have them write down what the one step is so they can solve it.  For instance, if they have an equation like x -3 = 8, they have to write down "You have to add three to both sides to isolate the variable." Or since I've got a negative three, I have to add three to both sides so I have the X alone."

For one class, I am not asking them to explain what they are doing because they are still learning the basics of multiplying decimals.  When they get to adding and subtracting fractions, they are going to start explaining how they found common denominators and created equivalent fractions so they can calculate the answer.

Although they hate doing this, I am seeing an improvement in their understanding of why they do certain things to solve problems.   It sometimes takes me a while to decide what the best way to ask them to explain what they are doing but its worth the time.

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

Money and Rounding

  I've noticed that students in this location have  tremendous difficulty in rounding money when shopping or deciding when the value should be rounded up or down.

I've noticed many people who shop at the local store tend to just grab food, watch the total as its all being rung up and if it goes over a certain amount, they add and take off food until the total matches the amount they can spend.

Most of my students do not know they should round the prices to make it easier to determine the total amount they are spending at the store.

So I send students to the store with a list of products they need to find the prices of.  I use this to put together an activity that requires them to round the prices so they can estimate how much they might spend.  Unfortunately, when they round anything, they carry out the math first and then round the answer.

I've also taken time to discuss why you might round something up or down, depending on who you are.  For instance, if you are a tax collector, you might round a taxpayer's amount up so you get a bit more but if you are paying money to someone, you might round it down so you pay less.  When I brought that up, the kids looked at me strangely because they've only been told the usual rule of 4 or under round down or 5 and above, round up.

I actually took time to show how if $32.5447 is rounded up to $32.55 when collecting it.  The $.0053 adds up to quite a bit of money when you are talking 100,000 people.  On the other hand, by rounding it down to $32.54 when paying someone money.  I showed they could save quite a bit if they had to pay 100,000 people. 

This is not a topic covered in elementary or middle school math.  I wish they'd take time to talk about it so students are exposed to it.  Sometimes, I think nuances are left out when teaching mathematics.  I don't know if they teach rounding of money in real life situations or are the situations contrived.

I try to provide real life situations and experiences for my students so they learn more because some of the situations are not as obvious living in the wilds of Alaska.

As usual let me know what you think.  Have a great day.

Math Games

  The other day, I discovered a web based math site whose games work on the iPad.  Its Math Games!

Too often I'll find a site but it won't work on my iPad because its java based.  It is frustrating when that happens because my students are disappointed.  Even if the site states the games will work on the iPad, I check it on an iPad.

I found this when I was looking for a game for my pre-algebra students could play to practice adding and subtracting integers.

The site is actually designed for grades K to 8 but much of the 7th and 8th grade material can be used to reinforce skills in high school pre-algebra or algebra classes. Although it was not actually a game with things to shoot, it did work my students hard and they enjoyed it. Even some of my students who do not normally work, did so.  One even asked if we were going to do this again today.

I had them practicing the adding and subtracting integers from grade 7 material.  The exercise had four levels with 10 questions for each level.  I told my students they had to get all 10 right before moving to the next level.  This insured they slowed down.  For several students, the activity clarified the rules and they were able to do it.

I like they have several ways to find material.  You can either go looking at grade levels or you can check the skills page.  The skills page has everything listed by topic such as addition, fractions, geometry, or number properties.  When you click on a skill, they list the appropriate exercise by grade level so its much easier to find.

Each problem is set up as a multiple choice with possible answers.  The program provides immediate feedback so they know if the problem is right or wrong and there is a progress button so they see how many problems they've done and which ones they got correct.

In addition, there is a virtual scratch pad a student can use to do the work.  When they click the scratch pad button, an opaque sheet covers it but you can still see the problem to work it.  Once they have the answer, they can click the get rid of the scratch pad and select the answer.

As I said earlier, each topic has multiple levels which take them from easy to hard.  If a student repeats a level, the problems are different each time, so they cannot write down the answers to use again.

I plan to make this a regular part of my instruction. 

Check it out.  Let me know what you think.

Creating Islamic Art

  Yesterday while researching muqarnas I discovered a great unit on Islamic art and geometric design put out by The Metropolitan Museum of Art.  It is ready to go and is perfect for a geometry unit.

It has several activities which require the use of a compass to construct the designs.  My state standards still requires that skill so something like this adds a real life element to the unit.

The first activity begins by having students create a drawing using 7 overlapping circles. It sounds easy but the circles have to be equidistant so as to be even. 

The second activity has students find shapes within the drawings from activity one.  They find there is a 6 pointed star, 12 pointed star, hexagons, and triangles all within the 7 overlapping circles.  The third activity again takes the rosette from the first activity to create triangular and hexagonal grids.

The fourth activity has students go from one circle to five overlapping circles used in the next couple of activities where students find 4 and 8 pointed stars and octagons. Activity 6 has students finding square grids from the circles.

Activities 7 to 10 look at finding patterns for triangular, diagonal, 5 and 7 overlapping circle grids. 
The eleventh activity is the one with relevance to yesterday's topic.  It has students create six and eight pointed stars out of a circle.  The finished product looks quite similar to the shapes used in the Introduction to Muqarnas video from yesterday.

Each activity has great directions and good drawings so its not hard to follow things and end up with a great finished product.  I also know that more and more people are relying on apps for geometric constructions because you can find free apps to do the job.  I have a few compasses in my class but mostly I rely on the apps myself because the physical compasses can easily break.  In addition, they cannot stab each other with an app.

Let me know what you think.  I'm off to try a new app I found that looks quite interesting and is free.

Muqarnas

  I suspect you saw the title and wondered about it?  I did when I stumbled across the topic in the latest Make magazine. 

The simplest way to describe muqarnas is they are three dimensional renderings of two dimensional geometric design.  In addition, there are two types - North African (Middle Eastern) or Iranian Style.  The School of Islamic Geometric Design has a short explanation on the differences between the two styles.

The idea behind muqarnas is that they smooth transitional zones between one area to the next.  The interesting thing about these is they can be made out of cement or wood.

This slideshare provides a great introduction to the topic.  It includes information on muqarnas themselves, the people who helped create them, types, and even the history of them. It is well done and shows lots of examples.

So where does one go to find instructions for use in your classroom.  This 13 minute is a great introduction to making one out of cardboard.  The creator takes people through the creation process showing everything step by step.  Unfortunately, the measurements are a bit vague in that he states its what he chose but he takes time to show everything in detail.

In addition, this 24 page pdf has greater detail with hand drawn patterns so a person can see how certain ideas are created. Some of the information appears in the slide share but the reason I recommend this one is there is a whole chapter beginning on page 11 which goes into great detail on going from design to the muqarna itself. 

It discusses how the arrows meet with shapes, the 5 basic rules to keep in mind when creating a muqarna, and even reading muqarna graphs and subgraphs.  Everything you need to create a unit in geometry.  I could not find any lessons already created but this is a fascinating topic. 

Let me know what you think.  I'd love to hear from people.  Thanks for reading.

 

Which One Does Not Belong #2

Which One Does Not Belong #1

Real Life Transversal Angles.

The other day I created this comic strip on transversal angles because my students struggled with understanding each one.  I wanted a way for students to really pay attention to the location of the angles.

They actually read it and put it in their notebooks.  The day after I gave it to them, I asked them to create real life situations for each type of transversal angles.  I stated they could also use the positions of buildings to convey the location of angles.

The assignment required a lot of thought by my students .  A few occasionally asked if they were on the right track.  One young man asked if the two angles he'd marked on a two story house were indeed corresponding angles.  I gave him a high five.

Another young man created a triangle with over lapping lines and marked vertical angles on it.  I asked what it represented in real life.  The young man looked me in the eye while explaining it was part of the Golden Gate Bridge where cables crossed over each other forming the angles. 

I drew a couple examples from other students on a piece of paper to share with everyone.  The first, a cross drawn by a young lady to show vertical angles.  I thought the creativity was great.  The second example is of monkey bars.  The young lady marked the alternate interior angles inside the monkey bars.

The final example came from another young lady who marked in boardwalks between the store and the church so she could show how the store and church were located where the alternate exterior angles are .  All the while they searched for ideas in their minds, they consulted the comic to make sure they had most things right.

The final activity took place this past Wednesday when I ran a short jeopardy type game.  I drew a picture with things on it like of two airplanes landing on two different runways so the planes might be in the corresponding angle position.

The kids had a blast.  They'd carefully check the comic again, argue before writing down their answers.  One group missed the first couple questions and cheered when they got the third question right.

I tried a different way of teaching this topic because the traditional ways didn't seem to work well.  I'm hoping this helps them remember the material in a more relevant way.  As always, let me know what you think.  Have a great day.

Changing Student Misconceptions.

  If you have taught math any length of time, you know students have misconceptions on certain topics.  Right now my pre-algebra class is struggling to get past ignoring minus signs  and  adding everything. 

I got desperate and created a flow chart to follow so they had to stop, look, and think about the signs before completing the operation. . 

Forming misconceptions is a normal part of learning but they can impede the learning process because many times students do not know they have incorrectly learned the material.  Secondly, any new learning filters through the established misconceptions.  Finally, misconceptions become so entrenched that its hard for them to be changed.

One of the best ways to identify student misconceptions is to cut back on lectures and increase student activities.  It is when students are working, their misunderstandings become apparent.  In addition, the best way to replace misconceptions is by changing from a teacher centered to student centered classroom.

Another highly recommended method for eliminating misconceptions is to regularly show problems containing the misconceptions so students can examine each problem, identify the misconception, and discuss why these are misconceptions.  Another name is error analysis so students learn to identify the error.

Unfortunately, many of the misconceptions students have when they hit high school have been with them since elementary school.  One of the standard ones has to do with the idea that if you multiply by 10, you add a zero at the end but that only works with whole numbers.  It does not work when you multiply a decimal by a 10.

I love using the error analysis method in class.  Often students have trouble identifying what is wrong due to their understanding of the material.  It takes several days of showing the same type of misconception for students to begin recognizing it.  It is important to work on eliminating their misconceptions so they do better overall in math.

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

Math Vocabulary

  Math vocabulary is something my students struggle with.  I try to integrate vocabulary into my classes by using it on a regular basis but I need to incorporate additional activities to keep their interest in learning it.

I know as a child, I hated writing vocabulary words 10 times each along with definitions.  It all became quite mechanical and I never learned to spell them.

I've used the Freyer method before but it isn't enough.  I need other activities designed to use the words more frequently.  So I looked and found some great ideas.  I like the idea of having a math word wall so students know the words they should be familiar with.  Once the word wall is up and students are up to speed, they can then:

1.  Play Pictionary where one student draws a picture of the word and the others guess it.  This works well if students can do a half decent job of drawing but for people whose drawing looks more like scribbles, it can become hysterical.

2. The I AM game.  A student gives the definition of a word beginning with I AM and students have to guess the correct word.  The person who guesses the word does the next I AM clue.

3. The memory game.  Place the words on one set of cards, definitions on a different set of cards or set it all up on the smart board.  Spread the cards out.  A student chooses two cards trying to match the word with its definition.

4.  What is missing.  The teacher removes a word from the word wall before class starts.  During class the teacher asks students to identify the missing word and its definition.

5.  Word of the day.  This works well for small groups of students.  Tell students the special word of the day and they have to listen for its use.  Every time it is used and someone notifies the teacher of its use, their team wins a point.  At the end of the class, the group with the most points is the winner.

6.  Have students create short videos for each word to show they know its meaning.

7.  Play math vocabulary bingo.  Create bingo cards with the needed vocabulary words.  Create the bingo calls using definitions.  You choose a definition, they find the word and cover it.  Once they have five in a row, or cover the whole page, they have a bingo or super bingo.

8. Inside/outside circle.  Each student is given a vocabulary word on an index card.  Students then write a definition and draw and example on the back of the card.   Then half the students form a circle facing outward while the other group form a circle around the other group of students so students face each other. Students give a definition and ask for the word or give the word and ask for the definition.  Then the outer group moves left while the inner one moves right for a new partner.

9. Make each student a Word Wizard by having them find the use of words outside of school in real life.  They have to provide proof such as hearing it on the news (date of broadcast) or finding it in the newspaper (bring article), or a family member used it (bring note.)

10. Have students create vocabulary cartoons using cartoons to create the picture for students to remember along with the definition, word, and cartoon caption.

Hope you enjoy these suggestions.  I plan to try a few later this week in class.  Let me know what you think.

Manufacturer's Suggested Retail Prices.

  The other day, I looked at a sale and the prices were based on the manufacturer's suggested retail prices (MSRP) to prove I would save a ton of money if I purchased anything.

The MSRP is particularly pervasive in the car market.  The MSRP is also known as the "sticker price" and must be displayed on all new vehicles per a 1958 law.  If it is not there, the dealer must provide it if asked.

Although this price is set by the manufacturer and displayed on all vehicles, there is no obligation for dealers to actually charge that price. They have the right to sell it for more or less than the MSRP.  The suggested price does not include taxes, license, and other extra charges. 

In addition, it is usually set higher so it looks like people are getting a deal when they pay less. If the MSRP is different among various dealers, it may be because some charge the cost of getting the vehicle to the destination as a separate charge or it may be included.  It depends. 

Students should also be aware businesses do not pay the MSRP.  It is the suggested selling price and most savvy purchasers know you are not going to pay the suggested retail price.  Many stores always sell items for less than the MSRP but use that price to prove you are getting a bargain.

Point of fact, I just purchased a video set for a series at $9.99 but the same page showed the list price (another name for MSRP) as $24.99 or I got a 60% discount.  Have students search various websites for certain items to see how the prices vary, even see if the list price is the same at all sites.  They could easily calculate the savings off list price and off the lowest price found.

What is nice is many stores such as Target, Walmart, Amazon, all have presences online so it is much easier to compare prices of various objects from the classroom. It no longer requires assigning students the job to go to the store to find the price.  Life is a bit easier.

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

Can a Dropped Object Have Rate,Time, and Distance?

  If you read my weekend entries, you'll notice I had two this past weekend, asking people to find the distance it took for a coin to fall from the top to the ground.

I did that because I realized a dropped object has rate, time, and distance just like a car or runner does.  The only difference really is one is horizontal while the other is vertical.

Now admittedly, when you drop an object, you have a couple more things to keep track of but the concept is roughly the same.  The difference lies in having to account for gravity at 9.8 m/sec/sec in the equations.

Its easy to show both. NASA has a lovely chart on free falling objects without air resistance.  Velocity = acceleration * time.  Acceleration is defined as 9.8m/sec/sec. Distance on the other hand is acceleration *time^2 all divided by 2. 

Unfortunately, we tend to only present problems which traditionally fall into the rate * time = distance.  I think its time to sneak a few of these type of problems into daily work.  Imagine also sneaking in Galileo because he really was the first person to experiment with finding out if two objects of unequal mass would fall at different rates. 

This was the prevailing scientific thought originating with Aristotle who believed the rate of falling was proportional to its mass.  In other words a 10 kg rock would fall 10 times faster than a 1 kg rock.  At the time Galileo was mocked but time proved his observations to be correct.

Some real life examples of free falling objects include cats jumping off of ledges, throwing a pizza up into the air when twirling it, leaves falling from trees, or my favorite those free fall rides at amusement parks.  They are the ones that look a bit like an elevator, go straight up to the top, then are released to fall, scaring the tar out of people before hitting bottom.  There is even a certain amount of free fall involved with people who jump out of airplanes.  They free fall until a certain height when they engage their shoots.

The example I experience most of the time has to do with children dropping rocks off the top of the hill, not realizing my house is at the bottom.  I know there are people there when the rocks hit my truck and dent it.  You should hear me scream up the hill at them.

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

Warm-up

Warm-up

Why are Spreadsheets So Important.

  I saw a tweet the other day in which the person indicated it is important students know how to use spreadsheets because most managers use them among others.  Its considered a 21st century skill.

Spreadsheets can play an important part in the mathematics classroom because spreadsheets allow calculations to be carried out faster, organize data and information, makes math more fun, and allows answers to "What if?" questions.

Its the last one that is today's focus because you can play with minor changes to see how each change effects the final results.  For instance, if you compare various offers from the bank, credit union, or the company financing to see which one is the best deal for buying a car.

The last time I purchased a car, I sat down and compared the offers to determine the best for me.  The cool thing about using a spreadsheet, one can figure see how a small change in the interest rate effects payments or final amount.  Students can even change the down payment to see how the final amounts are changed.

Adam Liss mentioned he used a spreadsheet to see how small differences in interest rates of mortgages can effect the total amount paid at the end 15 or 30 years.  Usually when one buys a house, a person has access to several mortgages.  A spreadsheet makes it easier to determine which one is best. In addition, if you are looking at pricing items in a business, its possible to see how changes in the cost of materials can change your profit margin. 

I found a lovely 32 page pdf on evaluating spreadsheet models filled with great information developing a good spreadsheet model before showing how to apply the technique to various situations such as finding the best break even point for a new pair of shoes and for modeling the multiple criteria decision making for a restaurant looking for deciding where to place a new restaurant.

Each example is done in detail following the suggested steps for creating a spreadsheet model.  At the end is a list of problems students could apply the steps to on their own so they can find an answer. I love the detail used because it makes it easier for students to follow the steps.

Check it out, play around, have fun.  Let me know what you think.  I love hearing from people and thanks to Adam Liss again for his suggestion.

Modeling Software.

The other day, I wrote about a school who used modeling software to calculate information in a real life situation.  I know my school district has budgetary issues so even if I wanted any of the most well known programs, the district couldn't afford any of them.

So I decided to check and see if there was free modeling software out there I could use in class.  I would love to give my students opportunities to use it.

First is Maxima, a computer algebra system which covers a wide variety of functions.  Although it was developed back in the 1960's at MIT, it has been updated on a regular basis and has an active community.  It runs on Windows, Unix, and MacOS10 along with a few other systems.  The program is able to plot functions in two and three dimensions while providing accurate results. 

Second is Scilab, similar to Mathlab but free. This program is a free and open software for numerical calculations which operates on Linux, Windows and MacOS 10. The program states it provides mathematical operations, data analysis, simulation, two and three dimensional visualization, optimisation, statistics, modeling, control system study, signal processing, application development, etc.


If you are interested in modeling complex systems using mechanical, electrical, electronic, hydraulic, thermal, control, electric power or process-oriented subcomponents, then Modelica is perfect because it is a open source free program designed to do precisely that. This programs comes with a 306 page documentation for version 3.4 released back in April.  There is a library filled with free and commercial components and funcitons.

Rather than list hundreds of other possibilities, I've included the link for a list of computer simulation software you can explore to your hearts desire.  Yes its a wiki list but as with any other list, some of the information may be wrong or out of date but the few I explored worked well and all of them sounded interesting.  Unfortunately, I do not have time to explore it all.

I think once the computer department gets caught up and organized, I'll download a couple things to play with before I have them put on any computers. Its hard to even discuss modeling in Math if we don't offer students the chance to use it.

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

Misleading Stats.

Yesterday, I ended up teaching a half day of school before taking students out for a 2.5 hour walk on the tundra.  I gave students a chance to catch up with their work, I looked for a unit on misleading statistics for students who were up to date.

While searching I found this wonderful document titled "How to lie, cheat, manipulate, and mislead using statistics and graphical displays" from UCSD.

This presentation is wonderful because it begins with definitions of statistics and such before moving on to explain different types of bias found in interpreting the data.  Each type of bias begins with a certain situation being given such as the recording temperatures by one buoy in the ocean around San Diego. 

The person takes time to explain the sample and the population parts of the situation.  It goes on to explain that if the data is applied to certain situations, it becomes a biased sampling and why it is called that.  From here, the presentation moves on to explain four types of biased sampling - area, self selection, leading question, and social desirability, each with the appropriate examples.

The next step is to look at the different types of data analysis used to manipulate information such as poor analysis, averages, and best of all, graphical displays which make manipulating data so much easier.  I love the way data is taken and using different ways of displaying graphically, people could come to the wrong conclusion.  The last couple of pages shares good graphical displays.

Combine this with a great article from statistics how to and you have a good introduction to the topic because it shows graphs which look wonderful but are totally misleading.  One example shows a newspaper comparing its circulation to another one.  At first glance, it appears the first one has double the circulation of the second but if you look at the actual scale, there is only a difference of about 40,000 readers or about 10%.

The misleading graphs are divided into missing the baseline, incomplete data, numbers not adding up correctly, two Y axis, and just reading it wrong, each with one or more examples.  It is great because the author of this included a written description of the problem and for one  included what the graph should look like.

Check both sites out and let me know what you think.  I had fun finding these and I plan to use them in class when we spend a couple weeks looking at statistics and probability.  I hope you enjoyed it as much as I.  Have a good day.

Setup Costs and Linear Equations

I am heading into linear equations in my Algebra I class.  As part of it, I include the usual business equations in the book.  Unfortunately, they assume that students will do the problems without understanding what the m and b are about or the students already have the knowledge.

Unfortunately, my students do not understand the basic premise enough to understand what the equation is about.  I do not want them to blindly plug in numbers because they don't understand the results.

So I have to include a lesson on what the slope and the y intercept represents within a business context.  I take time to establish  a scenario that they are soon to be business owners who have to determine the cost of making an item. 

This is fun because they have to determine the setup cost before they even start making the item.  Then they have to figure out the cost of making that item. The second step is to determine the revenue of the same item.  Finally they determine the the break even point.  This activity puts it more into a real world situation.

The one thing I have not been doing but will start this year (Thanks, Adam Liss) is having them use a spread sheet to see how adjustments in cost and setup can effect the final amount made and the breakeven points.  How does one penny more or less change everything.  They need to see this through the use of spread sheets.

The nice thing about spread sheets is they automatically change the graphs for you, all you do is change the amounts.  The visual graphs can make a greater impact on students than just the numbers because they see the changes by looking at just the slopes.

To prepare students for this, I found Khan Academy has a great section of instruction focused on linear function word problems that are actually decent as far as real world goes.  Fuel consumed, gym memberships, icebergs, etc. 

I think too often we teach linear functions with just the equations. We should take time to discuss the possible meanings behind the equations to give students a chance to see there is meaning.  I just had the students do a performance task involving two different linear equations and one student realized the slope was different. (Yeah.)

As I've said before, we need to add context to the equations so they understand the concepts better.

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

Happy Labor Day

Warm-up

Warm-up

Real World Calculus.

  I subscribe to the Math Education Smart Brief which publishes a few math related articles every day or so.  There are often articles I find extremely exciting such as the one in The Dalles Chronicle in Oregon.

The article talked about a group of high school students who studied calculus in a very nontraditional way.

They used exercise to learn calculus.  Exercise such as squats and vertical jumps are charted so they have data.  Then they consider 24 variables and constants like sleep quality, diet, etc, to see how they relate over time.  The idea is to calculate the maximum number of squats or vertical jumps at one go in their life time.

One student added additional variables like clothing and room conditions because a person can do more squats in leggings than jeans. If the room is too hot, it may impact the number a squats done.  Once these variables and constants are identified, the variables are ranked in importance in regard to the other variables and constants. Once they have assigned all those values, the information is put in a program and the results are calculated. They are able to calculate the final life time results and it can plot the life time progress on a graph. 

This gives students a real life application of calculus.  It often takes several tries to get the the data done correctly so they get results.  Its a great learning experience. The next planned project is to attempt to calculate what it would take to sustainably grow enough food locally for the community.  Such inspiration in finding ways for students to learn applied calculus.  I wish I'd had a chance to do this when I took Calculus but I got it via the old lecture and do lots of homework.

I'd love to hear what everyone thinks.  Yes, I realize today's entry is a bit shorter than normal but this seems to be a unique.  I didn't find any other schools doing this.  Have a good day.