Drones and Math

  I own a mini drone complete with camera.  I bought it for myself but it made me wonder if there is a way to use it in my classroom next year.  I would like to try to create a unit with the science teacher that integrates drones into both classes.

Right now, drones are becoming more and more useful in the business world.  They are not just toys.  I heard of one company who is working on using drones to deliver small things to remote areas.  For instance, using a drone to deliver medicine to a place cut off due to avalanches, etc.

There are also companies who use drones to map areas looking for damage to roads, etc.  Of course we all know the military uses them but more and more companies in the civilian sector so how can I as a math teacher use one in class to teach math?

The most obvious use is through the Rate x Time = Distance formula.  Well some middle school students in Elon NC are using drones in math class to calculate velocity using the Pythagorean theorem and the RTD formula.  According to the article, this activity grabbed student attention and it is opening their eyes to STEM.

The Academy of Aerospace and Engineering has a lovely discussion on using drones for STEM lessons.  In it, they discuss using math to find the wavelength of the drone's transmitter frequency because they know this particular drone broadcasts at 2.4 ghz.  This is one activity I had not thought of.  Cool.

According to Edudemic, you need a basic knowledge of trigonometry to design and operate drones.  There is also math involved in the electronics portion when building the drone.  What about using the coordinate system for the drone when it surveys land.  It has several other suggestions for uses but those are not as math oriented as the others.

It took a lot of digging but I finally found a pdf that focuses specifically on teaching math using drones.  Its only a 7 page file but it is geared for math classes with no adjustment.  The first part asks questions based on the information given so you can determine the flight plan.  The questions look at a variety of situations such as flying in calm weather, with cross winds, head or tail winds, calculating angles of correction and diagramming it all.  its a very well done activity and has everything I want.

Now I'm off to think about more ways I can use drones in Math.

Pure Math Vs. Applied Math

I came across a nice column by the Hechinger Report group on this topic.  They state that according to a report from Organization for Economic Cooperation and Development (OECD) finds that the way applied math is taught in the classroom is not good for the students.  It appears that lower socio economic students receive a watered down version of applied math while students from the higher socio economic groups are taught a more pure math.

Apparently, the difference in scores between 15 year old students who were exposed to more pure math tasks and those who were least exposed is about two years of education.  When this group looked at the PISA and compared it to the student's background they discovered that students who understand the concepts, they can make the jump  and apply it to other situations.

However, if students only learn the tricks,  shortcuts, and solving small everyday problems, they are unable to transfer their knowledge.  This comment right here may explain why the students I get in high school are not able to transfer their knowledge.  Their earlier math teachers teach the shortcuts, tricks and do not teach the concepts or foundations associated with the math.  Students need to learn the broad concepts, mathematical notation, and real world applications to fully succeed in math.

This particular report supplies an explanation on why students from wealthier backgrounds tend to do better in Math. It boils down to the math classes and expectations for the higher versus lower economic groups.  The students who receive the applied math focus more, are not exposed to the complex multi-step questions that require problem solving and significant thought. Furthermore, they learn to solve problems almost mechanically.

This is one reason Common Core focuses on requiring students to boost conceptual understanding.  For instance can a student explain why we when we divide a fraction by a fraction, we change it into a multiplication problem?  That is one reason many teachers ask students to illustrate the concept.  Aside from meeting the Common Core standard, it also increases student understanding of the concept.

I found this report to be both fascinating and supports something I've always been aware of.  I feel as if I spend half my time scaffolding many of my students who are missing chunks of knowledge due to any number of reasons.

I'd love to hear from others on what they think of this topic.  Thank you in advance.

Population Growth

  In the same chapter that has interest formulas, there are formulas for population growth and radio active decay.

Answering the question of why do we need to know half life of radio active elements is easy.  Since we have to dispose of the waste from power plants, x-rays, etc we need to know how long they need to be buried but when asked about population growth that can be a much harder thing.

So why do we need to be able to predict population growth?
We need to predict growth for:
1. Figuring out how much housing is needed for the additional population.
2. Deciding the number of cars that will be traveling the roads which determines how many new roads should be built or how many roads need to be expanded.
3. How many portables will be needed to added to the local schools for the increasing population or will new schools be needed? What about colleges?
4.  How much more money will be needed for the growing population in the over 65 sector?
5. How much more food needs to be grown to keep up with the increasing population?
6.  How much more demand for energy will the increased population require?

If students visited local census records, they could find enough points to calculate the growth of their city or country using a spread sheet.  Once the rate has been calculated, students can use that rate of growth to predict the population in 10, 20, 30, or 50 years.  Using the rate of new house construction, they can project if there will be enough housing available in 10 years for the predicted population.

This would create a connection between the theoretical and the real use of calculating population growth.

Which Car Offer Is Best?

  If you watch any amount of television, you will note the number of commercials that appear trying to interest the viewer in any of the newest car models.  Some cars are sleek and men magnets who want to own one.

You have your SUV from a variety of manufacturers showing everything from the onboard DVD system to the GPS and help systems.

You see some offers for leasing the vehicle and other to buy it but which offer is the best.  If you add into that offers from the bank or credit union, it becomes even more confusing.

All of this makes for a great project.

1.  Have students select a car that comes with an offer of so much off, lowered interest, etc.
2.  Find two loans with interest rates one from the bank, one from a credit union.
3.  Figure out three reasonable down payments.  Treat this as their first car.
4.  Find out the sales tax, licensing fees, etc charged by the state.
5.  Use a spreadsheet either numbers or excel.

So students use the spreadsheet to enter information from the car sales place and the three down payments to see how much interest they would pay on each over a 48, 60, and 72 month time period.  This means calculating it with a down of $2000 at 5% interest for 48, 60, and 72 months to  see how much they would pay.  Repeat for the other two down payments.

Repeat this exercise for each of the other two loans for the 48, 60, and 72 months to see the total paid.  Take the final amount of the loan plus interest and divide by the number of months to get a rough estimate for the monthly payment.

 Once this is done, choose one loan and try it with an interest rate that is 1/8 %, 1/4% lower and above the current rate to see how much a small difference in interest can make.

Then, have students go to http://www.cars.com/go/advice/financing/calc/loanCalc.jsp?mode=full or any other car payment calculator to see how much the monthly payments are for each of the above to see how close your monthly payment agrees with the actual monthly payment.

As a final step, have students write a paragraph explaining which loan is the best for buying the car.  The offer you see on the television is not always the best offer.  This also gives students a chance to apply their knowledge of interest

Distance Warm-up

Warm-up Question

Instead of having only written warm-ups, why not use pictures with the questions on them to create student interest?

Simple vs Compound Interest Part 2

  Yesterday, we looked at what types of loans used simple interest but when is compound interest used? A few things I looked at yesterday that I thought were compound interest turned out to be simple interest.

Compound interest is considered as profit growth, growth being the key word.   So this type of interest is applied to accounts you want to grow such as for retirement.  Depending on how much you invest and how long its invested, it will determine how much you accrue in your fund.


Money under 30 has a wonderful diagram and explanation of what saving a bit of money for a 10 year period at ages 25, 35, and 45 and letting it grow till retirement.  It is very clear and absolutely great.  I know from personal experience that at first it seems like your funds are not growing but after a few years, everything starts growing and it just becomes a larger amount than expected.

Another item that uses compound interest is your credit card.  If you only pay the minimum payment, it takes a very long time to pay it off and you pay way more interest than you might otherwise.  Business insider has some wonderful examples that demonstrate this using real life scenarios.   Credit cards break the per year rate down to a daily rate and recalculate the amount of interest based on this and an average balance. 

Basically if you are charged APR or or annual percentage rate it indicates they using simple interest but if its APY or Annual percentage yield, it indicates they are using compound interest.  In general if you are borrowing a set amount for a specific time period, its going to be simple interest or a variation of it.  If you are setting up something with a varying and ongoing amount like your credit card, its going use compound interest.

So the next time I teach these formulas, I can give students more information on when these sorts of interest are used.  I realize there are exceptions and variations but for students who don't know much about borrowing money, this is a good start.

Simple vs Compound Interest Part 1.

At various points during the school year, we teach the formulas for interest, both simple and compound but do we take time to expose students to real life uses of both?  Let's face it, most people receive all sorts of credit card offers. There are advertisements on television to borrow money for cars, borrow against the home equity for their hearts desire, even ads for so many months before interest kicks in.

Most people do not know when simple interest or compound interest is calculated.  I usually assume its compound but after a bit of research, simple interest is used in certain cases.

According to Investopedia - compound interest works better for the investor but simple interest  is better if you are the borrower.  For instance, simple interest is usually used for Certificate's of Deposit of one year or less because interest is usually per year on these. I can tell you that Certificate's of Deposit are running below 1% right now on this type of investment.

The majority of car loans are calculated using simple interest.  Its easy to determine how much interest you will pay at the end of the 5 to 7 year period.  Although it is a simple interest calculation, the amount of money paid against the principal is less because the payment is set up to pay more interest at the beginning.  By the end of the loan, more money is paid against the principal.  One major factor of how much interest paid on the loan is determined by the length.  The longer you borrow the money, the more interest you are going to owe.  The shorter time means you pay more per month but the total interest paid is going to be less.

I don't think my students realize this.  I am not sure how many students are fully aware of this.  In my neck of the woods, use snow machines or ATVs are a much better example because people don't usually buy cars out there.  It appears most personal loans are also based on simple interest calculations. 

Once you get to home loans, things get a bit more complex but are still considered simple interest but the complexity arises in the fact that you the principal payment varies.  This is another one where the interest amount paid is more at the beginning and the principal paid is more at the end of the loan period.

It appears that most loans want the majority of interest paid off first and the principal second.  This seems to be common for most loans when buying things but when do we use compound interest?  Check tomorrow for the answer to that.

The Break Even Point

  It always seems as if the idea of a break even point comes up in both Algebra I and II during the section on linear equations and systems of equations.  The examples seem to cover the hiring of two buses for a trip,  comparing cell phone plans, etc but many of these examples seem very unrealistic to my students.

So what are some ways to teach the topic so students see why we might want to know that information.  Saturday, I stopped through the AT&T phone store to get more information on cost of phones and contracts so we could figure out a breakeven point but it turns out they've changed the way they cell phones.

They now spread the cost of the phone over 24 to 36 months plus the monthly charges for usage.  So that now throws the old math problems out the window.  So what else can I use?  The other day I bought a magazine on start-ups your way and much of the information applied to franchises. Think how perfect a franchise is to determine the break even point?

As a teacher, I do teach the break even point when doing linear equations but my students have trouble with the basics of this particular topic because they live in a very small place that has two stores, one hamburger joint,  and a laundromat.  So why not use the information from the magazine in addition to using information from a few sites that provide information on fixed and variable costs.  In fact, students should be able to research the information needed to calculate the break even point.

The thing about the franchising fee itself is that it is often a flat fee plus a continued royalty which is a percent of income.  Entrepreneur magazine has a great article explaining  franchising fees that is well written and not too technical so it is good for English Language Learners.  The fees are a linear equation.  In addition, they same magazine has a nice article on calculating your break even point in a clear way, including examples.

Entrepreneur also has step by step  directions for calculating the break even point.  It is a five step process and each step is explained clearly with an appropriate example.  The examples are clear and easy to understand.  It is easy to find all the actual mathematical equations dealing with break even points but they don't always give the important information on what falls under each part.

Franchise know how is a site designed to help students determine what a fixed cost is or a variable cost.  It provides examples for each type of cost and even has a calculator to help. I would think that the cost of the franchise itself is part of the startup cost because its a one time fee. 

Yes, this type of activity might actually be a project or take time to complete.  I have one class a week that is only 40 min long.  I plan things like this for that one day.  I also have to provide step by step instructions since most of my students are classified ELL but these can be done.

Math Scribble App

Math Scribble App is an app with blanks that students can write on to demonstrate knowledge.  It had blanks for horizontal and vertical number lines (0 to 10), the first quadrant of the coordinate plane, horizontal and vertical number lines (-10 to 10) and full coordinate plane (-10 to 10), unlabeled horizontal and vertical number lines, 2 horizontal lines, Venn Diagram, Big Square,  10 x 10 square block, 10 rows of 10 blocks, 2 rectangles, 5 long rectangles, 4 circles and a totally blank page. 

A student can use their fingers to write on the page and if there is a mistake, its easy to erase and redo it.  Each of these pages can be used in different ways.  I took a few shots of some of the blanks and I can see a variety of ways to use them. 






This is the 5 large rectangles page.  It could easily be used to write down the correct order for solving an equation based on something shown on the board, or perhaps as a way to compare fractions, decimals?

 

This vertical number line could be used by students to add or subtract exponents, calculate thermometer readings,  elevation above or below sea level, etc

 

This particular page is good for showing the angles associated with two parallel lines and a transverse.

The blanks can be used in most high school math classes from Pre-Algebra up to Algebra II in a variety of ways.  In addition this app could be used in the upper elementary and middle school classes.  I plan to put this one on the iPads come fall because of the possible uses.  Check it out and the best thing about it?  Its free.

Crate World

Crate World is a free app for the iPad that is designed to help students develop math skills for number lines and coordinate planes.  It has five levels for students to work through beginning with simple number lines and ending with finding the equation of a line starting with two points.

Each level has multiple activities focused on a specific skill.  For instance level one or Apprentice one works on having students develop their skill using both horizontal and vertical number lines.  They work with both positive and negative numbers.  Occasionally, they throw in an absolute value.

The first few exercises have students locate the red crate x units from the green one both horizontally or vertically.  Other times, students are required to place a crate in a specific location or place it at the answer to a problem.  The last part of this level requires the student to find the distance between two crates.


Level two or Apprentice 2 are required to move crates on a 10  by 10 grid with numbers from -10 to 10 both horizontally and vertically. The level begins by having students identify the coordinates of a crate.  Then they must locate a point on the graph, identify the rise and run to get from one crate to another, move crates and then find distance.



Level three or Apprentice 3 begins by having student identify the equation for a vertical or horizontal line. Moving on to identifying slope, locating a crate on a point of the line given so the line visually appears, identifying the equation for a given line, finishing with solving for the horizontal or vertical line.

Level four or Apprentice 4 has students continuing with what they've done but they are solving for missing numbers in a variety of situations.  They have to solve for y, b, x, and m.  Each one is a different activity and the level culminates in finding the slope between two crates.

Level five or Apprentice 5 takes it all one step further.  It has students learn to rewrite equations from standard form to slope - intercept form, square of numbers, square roots, distance using the distance formula, finding slope from two points, and determining which set of coordinates will make the equation true.

If the answer is not correct, the student receives immediate feedback by either having the crate move to the next button to try again or it will say try the next problem.  The app is free and provides multiple opportunities for students to practice all aspects of finding and using linear equations. 

I like the way this is set up more as a game than an instructional instrument.  I also like the way the first level is predominately focused on helping students learn to use horizontal and vertical number lines.  Many of my students are weak in that area and I can get them playing it much earlier in the semester.  I look forward to using it in class this fall.

Happy Fathers Day

Possible Introduction?

Simple picture taken at the local rocket range.  Good for an introduction in elementary class.

The Mathematics of Tesselations

  At some point, we teach students to create tessellations during the school year.  I know I have.  I've had students create either a concave or convex shape and use that to create the final tessellation but I have not taken time to discuss the mathematics of tessellations.  Have you?

How do you go about teaching the mathematics behind tessellations?  What things do you need to focus on?  What are the different types of tessellations?  What prior knowledge does a student need before starting?

This 29 page pdf answers many of those questions and provides some excellent information for creating a short unit on the mathematics. It begins with explaining the knowledge students should have when beginning this unit and reviews the formulas to find the interior angles of any regular polygon.

Once it gets into tessellations, it has students explore creating tessellations using one regular polygon.  Its interesting that there are only three regular polygons that tessellate but students are expected to discover this by trying the different polygons.  The author provides the mathematics to determine mathematically if a regular polygon will tessellate.

It goes on to discuss other tessellations in detail in such a way as to show the consistency of mathematics involved.  The author includes Eshers tessellations and even creating tessellations using irregular shapes.  Each type of tessellation is detailed with lots of clear examples.   This article takes the information from the previous pdf and provides a great recap of the material.

Now for the big question!  How are tessellations used in real life?  Are they?  Yes, they are.  This web page has wonderful examples of how tessellations are used in real life.  The most obvious example is in a bee hive because every cell is hexagonal shaped.  If you notice, hexagonal is one of three shapes that tessellates using one shape.  Other examples include chicken wire but if you've ever seen a fishing net, it also tessellates a regular polygon.  What about quilts, floor tiles, brick work, or certain toys, reptile skins, decorative wire works? 


Its just a short step to having students create a final product for tessellations that might be:
a.  A presentation.
b.  A pod cast.
c. A video.
That covers :
a. Tessellations.
b. Real world uses.
c. Artists who made their name using tessellations.

So many possibilities just like tessellations.

Pythagorea

Front page

 The other day I searched Geometry apps that could be used in class.  Unfortunately most of the apps I found were either games or made for early elementary.

Occasionally, I found an app that could be used to either reinforce learning or used in a flipped classroom but they were far and few in between.

I stumbled across Pythagorea: Geometry on Squared Paper which is a Geometry app that tests student understanding of certain geometric topics by having them construct it.

Levels

 Rather than focus on individual topics, it reviews a variety of topics in each level. The topics range from squares, parallelograms, perpendicular bisectors, midpoints, angle bisectors, etc.

Although the first level is labeled elementary, it really is a way for anyone to become with familiar using the app.  Each level has at least 10 problems and you must complete each problem before on to the next.

Furthermore, the student must complete all the problems in a level before moving on.

Problem from Very Easy Level

What I find most interesting about this app is that it does not provide the correct answer if a student is wrong.  If an answer is correct, it flashes yellow at the top about the level and problem number while the right arrow on the problem turns red.  If the answer is wrong, nothing happens and the student is expected to keep trying until its right.

In addition to learning what each item is, students are required to develop vocabulary so they know the terms and definitions.  I played the first two levels of this and there were a few questions that made me think and I had to try two or three times.  

Once a level is finished, the app creates a fractal tree based on squares and triangles.  You never know what its going to look like until you are finished.

You do have to have a basic knowledge of geometry to use this app but it would be possible to have students use this with a book so they can look up the material as they use it, otherwise use it once students are at least one quarter into the year.

I like this and I enjoy playing it myself.  I love the fractal trees formed at the end.

Scales and Maps

Did you know there are three types of map scales?  I didn't until I found this article from the San Francisco Estuary Institute and Aquatic Center.  I knew about the first two but the third one?  I'd never heard of.  I usually teach a little bit about scaling during the unit on ratios but I'd never done much with maps.


The three types are verbal, graphic or bar,  and RF or representative fraction scale.  They give great examples, talk about converting from one to another type, and include information for calculating the scale from a picture or map.  This is a wonderful real life application of ratios and scale.  They even talk about flat vs non flat areas when calculating scale.

Furthermore, this article goes on to discuss finding area and distance using both the map and the scale.  I love the fact they take time to discuss four different types of measurement used in the field and even go so far as to discuss accuracy, precision, and significant digits.  Although it is just an informational article, it is filled with lots of examples.

So with this information, you could use a prepared worksheet such as this one that has you estimate and then calculate distance on a map that has a only one type of scale.  It wouldn't be hard to add another page where they have to convert to the other two scales. It might even be possible to find something in Google Streetview that the students could use to practice calculating distance based on the picture you are looking at.

What about an online game that allows students to practice measuring distance on a map and then calculating the actual distance using a world map? Perhaps you could add a worksheet to this so the students write down starting, finishing, distance measured, actual distance so once they have done say 10 locations, they can calculate the predicted time it might take to fly between the points based on average airplane speeds.  This is a real life application for map scales and rate x time = distance. 

More tomorrow on this topic.

Curious Question on Fractions

In math, we teach that a fraction represents an part of a whole. The part may be distance, part of a total, etc., but it is always an equal part.  I was walking over to the library today and realized that we sometimes divide distances into segments based on landmarks rather than actual distance.

This may be one reason some students have trouble with drawing pictures showing fractions.  I had a student several years ago who would draw a rectangle and divide it into three uneven sections.  She honestly did not know the segments were supposed to be even.

Think about the different ways we use fractions including estimation.  I did a search to find out why we estimate fractions and how estimation of fractions is used in real life but I could not find much on the topic.  Most of the material I found is on how to estimate, figuring out if a fraction is closer to 0, 1/2, or 1 but nothing on its real world uses.

So does that mean it is not something that is done in real life or is it not considered important?  I know that when I buy things by the pound I might say "I'd like a quarter pound of tea.  Please get as close as you can."  Or the sales person might say "You are just under 1/4 pound."  These are close but not exact. In fact, sometimes, places break the price down to a per ounce weight rather than deal with fractions of a pound.  Many tea shops price tea by the ounce since that is easier to use.  

After a bit more search, I found a few things on how to estimate fractions so you are actually founding the fraction to something a bit easier to use. An example would be 2 1/3 is rounded to 2 for estimating a total sum.  The example here was estimating so you'd know about how much ribbon to buy.  That does work but you'd never round down for ribbon because you want to make sure you get enough.

When you sew a dress or anything else, you tend to round the amount up so you have enough material for the outfit just in case you make a mistake.  But that does not always answer the question "Why do I need to round fractions?"

I would love it if people could answer the question "What are some real life examples for rounding fractions?"  In other words, why would we estimate fractions in real life?"  I could use some help.  Thanks in advance.

Color Theory and Math

  I was actually out searching for information on using 360 VR in the Math classroom and stumbled across this article talking about a math based approach to color theory.  It was something I'd never thought about before.

The application is specifically for Photo Shop but the math is facinating because it relies on a formula using 180 and 360 degrees.  The degrees refer to how far off the base color you are going.

For instance the picture only represents 180 degrees.  So the Yellow might represent 120 degrees from the red color on the half color wheel showing.  An application I had never thought about.  I've only through about degrees in a more scientific or geometric application.

The author defines hue, saturation, and brightness and gives the starting values within Photo Shop.  It is awesome to see that the basic formula is the absolute value of the (base color + so many degree) - 360 degrees.  If you are not sure precisely how certain types of color groupings work, check this site because it has wonderful illustrations that clarify types.  Apparently, Sir Isaac Newton created the first color wheel.

This article actually clarifies the creation of certain colors that are referred to in the article in the next paragraph and its done in a clear manner.  He talks about cyan as being white - red or what is left after the white is taken out of the red.  Although it is more about adding or subtracting color, it gives students a feel for the idea of the concept of taking things away or putting more in as an application, rather than focusing solely on numbers. 

This article talks about the math behind mixing colors but that it is not scientifically proven yet. The article is quite detailed and appears to be more for artists who want to create their own colors but it sounds fascinating and possibly might fit under ratios.  I like the illustrations they use to support their thoughts.

This tangent is rather fascinating to me because its  more of a conceptual application rather than always being a numerical application.  Let me know what you think.

360 Virtual Reality or Panoramic Views.

  Earlier in the week, I attended the Kamehameha schools Educational conference.  I enjoy going there because they always seem to be on the cutting edge of it.  This year, Michael Fricano had some excellent sessions on 360 Virtual Reality starting with what is it to how do you create your own.

I had so much fun learning about it but being the math person I am, my mind kept trying to find ways to use it in my classroom.  Last night, I figured out one way to do it.

There are apps created that use your phone to record the 360 photo.  A free one, he recommends is Google StreetView but my cell phone is still rather primitive only because I live in a place where service is not available.  So I didn't invest in a fancy one.  Back to the topic. 

Once you've recorded the scenery, it is possible to add annotations and links to your 360 picture using a different software.  He showed several examples of 360 in action with links and additions.  They were so cool but it was the one from the Spanish Class that set my mind off at a gallop.

The Spanish teacher had her students surrounding her when she took the 360.  Then she linked each student to a 360 they made of their rooms at home where they spoke about their room in Spanish.  Just think how it would be if a math student found a place in town they could take a 360 of and then annotate it or create links to the shapes they've identified in the picture.

These are a few way's I've thought of for using 360 in Math.

1.  Shapes
2.  Geometry vocabulary
3.  Various angles.
4.  Linear graphs.
5.  A Graph of Systems of Equations.
6.  Rate of Change/slope/pitch.

I haven't tried it yet but once I get home again, I want to try to create some math 360 videos or panoramic views to see how well it would work.  Often, I get the ideas and then I try them out because I want to have a variety of tools available in my math classes that allow students to combine creativity with mathematics. 

I'll probably provide an update later, once I've had a chance to play.  If you get any ideas, leave a comment.

  I love using tangrams in my geometry classes but I hate having to deal with all those little pieces that get lost or broken.  Unfortunately, most of the tangram program's I've found are all designed for the user to complete a predetermined design.

After some searching I discovered a web based tangram that gives you a choice of filling in a creature or creating your own.  It does not appear to work on the iPad but it does work on my Mac. 

The Tangram Activity is a bit clunky but it does exactly what I want it to.  I have a couple activities that require students to create certain geometric shapes from certain tangrams.  One figure is a triangle like the one above that uses all the pieces.  Others might ask for a concave pentagon using pieces 2, 3, 5, and 6.  This activity has challenged my ELL students in the past.  I don't know if using a computer based version is better than the actual ones but at least they can't throw them out.

I found a second one that was originally created in 1999.  Again, it works on the computer but not on the iPad.  It does allow a person to move pieces around to create the geometric shapes without  being limited. 

I actually found a bit more information on ways to use tangrams in high school math.  Starting with Grandfather Tang’s Tangrams Go to High School (you'll have to do a search for it because it was a word document).  Its a 9 page lesson that uses tangrams to teach area and perimeter.  The activity begins by having students figure out the area of the shapes before asking students to find the perimeter of certain creations.  I like that this is a true exploration where students have to deduce much of the information themselves.

This site has excerpts of Grandfather Tang's story to read to the students before having them use the tangrams to create a variety of quadrilaterals from rectangles to rhombus to trapezoids.  Some shapes can be created while others cannot.  In addition, students can classify angles within the various pieces which provides a great review.

Finally are the two lessons found at Tom's Math Lessons over at the Math Forum which have students trace, classify each piece, and explain how they found the area.  The lessons are well done.  They come complete with objectives, materials, the lesson and extensions.  In addition, there are answer keys for both activities.

I plan to add the lessons to my repertoire because these can be done as hands on and have a geometric perspective rather than just recreating pictures.  I love finding materials such as these.

Update on Paper Circuits in Math

The zeros in a polynomial

I had the privilege to present at the Kamehameha Schools Educational conference yesterday.  It was Awesome watching adults figure out how to create a circuit so they could use it in their classroom.  I provided several examples for the first half hour and for the second half hour, I provided people a chance to explore creating their own circuits.  Several of the elementary teachers figured out how they could use it in their classes.

Ellipse

 This type of circuitry works well in conic sections.  I used the same circuit for the ellipse and the hyperbola since the two equations had the same a and b.  It worked well.

Hyperbola

 Of course you can use lights to show select points in a linear equation. 

Linear Equations

 The LED"s show the point of intersection for the two equations.  It makes it quite clear what is happening with the graphs.

Systems of Equations

Since I work with ELL students, I've found using the lights help the students see the points so much better than without.  The leds cause those points to jump out.

These are just ideas I've played with.  I can use it for discontinuities.  A kindergarten teacher realized she could use the circuits for the shapes with her little ones.  I felt thrilled a couple of the elementary teachers figured out their own uses for paper circuits in their classrooms because I have no idea what would be appropriate for them.

Projectile Trajectories

  When you start talking about trajectories, people automatically think about rockets.  They talk about having to plot the trajectory of a rocket but what other things can we bring to the discussion on this topic.

The starting point would be to explore the multiple meanings of trajectory because it is easy to confuse at least two of the meanings.  I tried looking under trajectories and learning trajectories and could not find exactly what I wanted.  It wasn't until I looked at Projectile Trajectories that I found exactly what I wanted.

So how do you use projectile motion to teach math?  Well in most cases, even with bullets, the path is all or part of a parabola.  The Physics Classroom has a wonderful list of interactive simulations that can be used in the classroom starting with its own simulator.  Their simulator allows you to trace the parabolic path of the projectile, read the x and y displacement, and the velocity vectors.  It is easy to stop and start the path so a student can find the max height and displacement so they can calculate the formula for the path.  In addition, the speed, starting height, and launch angle can be adjusted according to needs.  Finally, this activity comes with a 4 page worksheet that turns this activity into a real exploration.

Out where I live, we have tons of hunters.  It would be easy to teach a lesson on bullet trajectories and possibly ways of finding the equations.  I do not know anything about using guns at all so I'm off to the internet to find information.  Well it turns out there is a site that has quite a lot of information including this site which has information on the Mathematics for Precision Shooters.  I never realized there were so many different formulas involved in shooting.  I thought you just aimed and shot. The  site is for serious shooters who may be in a competition or perhaps in the military.  It is mind boggling. 

The last item to look at is chucking a Pumpkin which is serious business for some people.  This site has an article where the author discusses how he came up with his values in order to calculate the height of a pumpkin in a toss.  I love the way the author goes through each and every value.  It gives the students a chance of understanding everything involved.  Check it out.

Quadratics - What Do They Model?

  Have you every wondered what the a, b, and c do in the quadratic equation?  What are they other than coefficients?  Personally, I never actually gave it a thought.  I had to identify them so I could use them in the quadratic equation to find the roots.  When I studied math, that was never a question that came up.  We accepted it.

I've been exploring the idea that quadratics can represent area but during my search to find more information, I stumbled across this Geogebra applet that is awesome.   It allows students to play with the values of a, b, and c individually so they can see what happens as each coefficient is changed.  I admit, I had so much fun playing with the applet that I had to show it to others.

The thing is, I had a really hard time finding the information I was looking for.  After a nice long search, I found that Plus Maths organization has a list of 101 uses for quadratics which has lots of details to explain the formula, who came up with it, and what it represents.  The information has been spread out between two parts.

Part 1 begins back in Babylonia with their record keeping on tablets.  The first example is simply the area of a farm and what happens when you double the length of the side.  Then the author moves on to the Greeks,  the Pythagorean Theorem, and the Golden Ratio.  From here, the Greeks and conic sections are discussed and includes illustrations showing how each conic section is cut out of the original cone.

Part 2 continues the history lesson but brings it up to modern times with more modern applications. It begins with a discussion of Galileo and his accomplishments.  The connection between his discoveries and acceleration or stopping distance both of which use the quadratic.  From there the author shows the connection to ballistics.   Then as far as historical references, they moved on to Newton, his contributions and his works relation to quadratics.  They moved to Bernoulli and air travel.  The last two topics include chaos and mobile phones.

Every example includes the mathematics involved and explains why and how its a quadratic.  It would be easy to take the information out of this set of articles to create lessons in class or assign students to work on a presentation using the information provided.

PUMAS

  No, these PUMAS are not members of the cat family.  PUMAS stands for Practical Uses of Math and Science and is actually the on line journal of math and science examples for pre-college education.  It is put out by NASA and has 88 different examples available.  The site has the examples that teachers can use to create better lessons.

Although there are examples for grades K through 12, the majority are for middle school and high school.   After looking at several of the lessons, I am impressed because many of these have combined science with math so neither is taught in isolation.  The material included offers so many different examples of math used in real life that it could easily enliven the lesson.

One middle school lesson has students creating a cross section from a topographic map and once the cross section is complete, students are requested to calculate the slope associated to the drawing.  The directions are clear and easy to follow.  The lesson is peppered with illustrations to show what the final product is.

Another lesson focuses on what happens when a ship carrying fuel becomes grounded during its voyage.  The ship carries fuel and the lesson has students perform the calculations converting from gallons to meters so they get an idea of how much space the fuel takes up.

The high school section has some great lessons such as one which looks at analyzing the statistical accuracy of medical tests for positive, negative, and false positives.  The lesson takes students through figuring out the probabilities for each condition and includes the appropriate graphs to illustrate the math.  The explanation is well done and is quite clear.

Another exercise looks at snowmelt and flooding which is an important topic in many areas.  It builds prior knowledge by discussing a film many students will have seen.  It requires students to calculate the volume of the snow in a basin.  It also provides the solution so the teacher knows how it was done. In addition, there are questions that are somewhat open ended as to why the flooding may have occurred at a certain time.

It looks like most of the material was created back in 1998 or 1999 but its still quite valid and useful.  Check it out and I'm sure you'll find something that you can use. 

Parabolic Shapes

   I've been seeing references to teaching mathematics in context so students can see the relevance of the material.  I admit, I'm one of those people who don't mind learning things from the book that is taught almost in isolation.  I also admit, it is much easier not bothering to find real life situational contexts for any math topic I'm teaching.  There are children out there who need to see how the topic fits into the real world and do not want to learn it if they have no "reason" to do so.

The nice thing about parabolas is that it is easy to find real life examples that most students have seen.  Out where I am, the most common application of the parabolic shape is seen in the satellite or microwave dishes.  We have both scattered around the village.

In addition, most vehicle headlights use the parabolic shapes due to the ability it gives to focus the beams, making the lights stronger and easier to see.  Another use for the parabolic shape is with certain types of skies.  These cuts make it easier to turn on the skies.  The water in fountains be they decorations or drinking, always form parabolas.

It turns out there are so many more uses of parabolas and parabolic shapes.  Did you know that breaking distance and total stopping distance formulas are quadratics and form parabolas when graphed?  Certain types of heaters make use of the parabolic shape to distribute the heat out into the room.  The cables on the Golden Gate bridge are in a parabola.  Other parabolas include microphones, mirrors, ball throwing, a flight creating weightlessness, trajectories, basketball, vertical curves for roads, overpass designs, and so many more examples.

Wouldn't it be so cool if we provided pictures of parabolic shapes with some basic information so students could create the equations for the object.  Image showing a picture of the Golden Gate Bridge and giving students the height for the top of the posts for the suspension part of the bridge and the height at its lowest point between the posts on the bridge.  That should be enough to calculate the formula. 

It might not be a bridge, it might be the formula for a 3 point basket thrown from a certain point on the court.  What about figuring out the height of an entry in a pumpkin toss based on its final distance.  These activities could put the math into a context students relate to instead of just plugging in numbers and creating equations in isolation, without the context. 

Yes its hard.  Yes it is going to take some work but if it helps students learn the material better, then its worth the time.  If you have anything to contribute, please do so in the comments section.

Why Is Slope Important?

  I always have that one student in school who wants to know why it is important to learn about slope.  After all, we never ever use it so why do we have to learn it.

I always ask them why they post the grade on steep hills in the mountains?  Why do truckers need to know that they are about to head down a road that is 6% grade?  That usually has students stopping to think.  Unfortunately, it doesn't always work with students who have very little experience driving down a real road.

The use of slope, grade, or pitch is a wonderful way to introduce the rate of change.  Instead of saying the mountain has a 6% grade, put that in actual circumstances such as if you travel 100 feet, you will go down 6 feet.  That gives it the real world context that is often missing when you discuss slope.  Slope and rate of change need to be discussed together to establish the relationship.

This article "Making Sense of Slope" has lots of great real world applications that show a variety of  contexts of slope.  It has everything from the cost of a watermelon, rate of earnings based on a per hour rate, growth of cable television, etc.  The article is very detailed with diagrams to show the rate of change.  It even addresses the question of a rate of change that is not constant.

This site has a lovely 35 page packet including the slides with great real world information including information on different types roofs with links to sites that explain how to calculate pitch.  It includes information on road grades for roads, curbs, ditches, wheelchair ramps and it brings in the Americans with disabilities act.  In addition, it addresses avalanches, skiing, and related topics.  At the end, there is a great worksheet for people to calculate rise over run for a specific gradient.

I plan to take a week to assign students a chance to create a presentation such as a video or slideshow to show real life examples of rate of change.  They can use examples found around town such as the microwave tower for undefined slope, or the foundation of someone's house for a zero slope.  It will require them to apply their knowledge of slopes to create the presentation.

Free Calculus Apps

Today, I thought I'd do a check to see what apps are available to use with Calculus.  I have one calculus student this semester so I need to know a few apps I can have him use.  I tend to go for free apps for the simple reason I hate to pay for an app that does not do what it advertises.

Next is Magoosh Calculus Lessons which uses animated videos to teach differentiation and integrals.  It has over 6 hours of videos to watch on three topics, precalculus, derivatives, and integrals. Each video is between two and three minutes long.

 Magoosh does require you sign in but its a free sign-up and takes two seconds to do.  All I had to do was enter my e-mail, a password, and I was on. 

The videos are listed by topic with subtopics for every topic listed so you can choose exactly what you want.   Although it is designed for calculus, it could easily be used in other upper level math classes because it has videos on trigonometric identities, even and odd functions, transformations of functions and composition of functions.  This app could easily be used in several math classes.  You do need an internet connection to watch the videos but you can watch the videos as often as needed.


Mathtoons has an Intro to Calculus app which is free and is a great way for students to check their knowledge.

This app quizzes three important areas in calculus.  It reviews functions, tangents and secants and limits.   Since it focuses on introductory calculus, this has three important topics.


 Each topic has 10 quizzes which test one facet of the topic.  When you click on the quiz option, it tells you what it tests and the difficulty level.

I clicked on quiz 1 which is a function review with a difficulty of two.  It tells you that the math in this quiz should be done by students without using a calculator.  I took the quiz and the questions primarily asked about alternative forms or which was more.  I purposely missed a question or two  to see what would happen.

You get the question wrong, a note comes up giving some information and returns you to the page you were on to try again. Once you select the correct answer, you move on but you do not get credit for the problem because you missed it.  The app refuses to move forward until you get it right.

At the end of 10 questions, you receive a score. You can retake the quiz to redo it just don't mark down the answers because although the questions are the same, the answers have been mixed up. 

Bedtime Math

  I know, I know, the title makes it sound like a small child's app but that is not necessarily true.  The Bedtime Math App is based on the idea that you read to your children every night so why not read a short story and answer a question about the math.  I will admit that I'm not sure what ages this particular app is geared for but I do see a use for it in some of the lower level math classes and to help ELL students develop their vocabulary.

These stories are not the standard word problems you see in text books.  They are short stories that cover a variety of topics from ant hills to what it takes to be Superman.  The stores are detailed and some even have accompanying videos to illustrate the topic. 

 The stories start by trying to build on prior knowledge.  For instance, this one begins by mentioning super heroes most students know.  It gives a bit of information on when Superman first showed up and goes on to quote the first few words of the show.

The story then goes on to explain what the strength of a locomotive is, how fast is a speeding bullet and how tall can a building be.  So it takes the description and puts it into mathematical numbers a student can relate to.

Then they have the choice for the level of kid.  I looked at the big kids question which you see to the right. The question gives the height of the highest flying bird and then asks how high can Superman fly if he flies 10,000 feet higher than the bird. 

Once you have figured out the solution, you can check your answer at which time you are given a bonus question about commercial planes in reference to Superman.

There is always one extra question under "The sky's the limit" which is more complex than the earlier ones. In addition, when you check your answer, the answer will show the work if there are several operations involved in calculating the answers.

I like this app myself because I love learning new things and these problems are written so they are interesting and a person can easily relate to these.  I think I'm going to put this app on my classroom set because the app encourages literacy in the Math classroom.  Check it out.

Why Do We Push Students Into Algebra So Early.

  This past year, the whole middle school and high school has had to make significant changes due to the common core standards.   They've been around for a while but my district decided to wait until the last minute to implement them.

So now any 7th grader who is deemed ready based on the 6th grade teacher's recommendation will be taking Pre-algebra while the others take a basic 7th grade math class.  All the 8th graders will be in Pre-Algebra or Algebra. 

By the time they hit 9th grade, they are expected to be in Algebra I or Geometry.  I know that not all 9th graders will be ready to take Algebra I as some struggle with the basics. I teach where there is only one math teacher in high school (me) and two middle school math teachers for grades 6 through 8.

We know that all students develop at different rates and does brain theory support pushing kids ahead in math at faster rate?  I feel as if we've forgotten the student in the rush to push them ahead.  I had students who have reached the 9th grade who still struggle with signed numbers, fractions, and other basics.  If they are struggling, I end up having to scaffold them but they don't always develop the foundation they need because they are still confused.

Yes, I have a couple of students coming up who can jump into a higher math when they start 9th grade but most of my students still struggle with the basics.  We have elementary, middle school, and high school all in one building so I have input into what is taught in the lower grades.  There have been two elementary teachers who have a solid math background and have been working hard to help students develop a solid base but the others?  That is a whole different story.

I do worry about many of these students who are ELL or learning disabled being pushed faster than they are able to move.  I have a senior who has a print disability and he struggles in Pre-Algebra.  As he moves up to Algebra, it will be much harder because he cannot run everything through on a regular calculator.  My other learning disabled students do well if they can run the numbers on the calculator but one they reach Algebra or higher, I have to make sure they have more complex calculators so they can complete the work.   Unfortunately, many of these students still struggle. 

I will always worry about pushing these students forward before they move on because if they struggle with math, they often develop a dislike for the subject, convinced they are "unable" to get it.  When in reality, the system has decided that they need to be at this point even if they are not ready for it.  I have no answers but I have lots of concerns and the only thing I can do is teach the material as well as I can and provide differentiation and scaffolding and hope they do not develop a dislike of the subject or a closed mind.