The Math of Triangulation

I think most people have seen a television show or movie where someone has to triangulate a cell phone, a missing hiker, or even an earthquake but what is that exactly?  How do you triangulate?  Whats the math used in it.

Triangulation is defined as a way to use triangles find the source of an earthquake or spacecraft.Although there are several types of triangulation, we are concerned with the type dealing with distance or physical location.

One application is to find the distance of a ship from the shore using trigonometry.  In ancient times, two observers stood on shore a set distance apart.  They then used their instruments to find the angle of the ship and using trig, found the distance.  Maple Soft provided this example.  If you want to check out their interactive activity associated with this, please do. 

I found a lesson on Math Forum which takes the above idea into more depth but instead of using ships, it uses general items placed at certain locations but it takes students through the process of using triangulation to find distances.

Now this example talks about finding out the location of artillery (cannon)  using three towers or microphones using the speed of sound to find the distance.  Its very well written and is actually the answer to a person's question on this topic.  I love the way its explores  a real life situation.

Here is another one on the mathematics behind laser triangulation.  The answer uses the Angle Side Angle theorem to help answer the question.  It is beautifully simple and cool. 

These are just a few instances of triangulation and their math.  Check it out and enjoy.

Changing Teaching Style

  I've noticed that my teaching has changed quite a lot already this year from even last year.  I am finally getting it to where I feel like I'm reaching the needs of all my students.

Several years ago, I took training which focused on ways to help students learn better.  The training said you needed to expose students to the material over a 21 day period for the best chance of them remembering it well.

I kept trying to figure out ways to do this but kept retreating to my standard teach it and move on.  This year, I'm finally getting it together to do it the way I want.  My lessons look something like this:

1.  Warm-up for the first few minutes of class while I take roll, get materials in place, etc.  You know those little tasks that you can't do till class starts.  It also gives students a chance to settle down and focus on the work.

2.  I introduce something new such as the next step in the series such as identifying the intervals for increasing and decreasing of a function or polynomial.  I might show a short video or give a few introductory notes.

3.  Review the material we've been working on and either work examples, start the problem and have them finish or give them an assignment.  In other words, I do the work, we do the work, and then they do the work. 

4.  I usually spend the last 10 to 15 minutes working on skills they are weak in such as solving one step equations, rewriting a standard equation into the point slope form, etc.  I have apps on the ipad I have students use.  I have each student work on their own weakness.

5.  I finish the last minute or two of class with a verbal this is where we'll be going in our next class meeting. 

I find that step 4 is great because it allows me to keep an eye on their weaknesses and helps me diagnose other problems.  I just found out one of my 9th graders has trouble with simple arithmetic operations so I know what he needs to work on.

I don't always do it in the exact order as listed because things happen.  Sometimes I mix them up so students are not able to get used to the same thing every day but the general routine still works for them. 

The next thing for my ELL students is to look at their notes and examples so they can figure things out rather than always asking me "What do I do next?"  This particular trait is not only done in my class, it shows up in other classes. 

I like the way this works and each section does not cause students to be bored or overwhelmed.  It also makes the class go faster.

Rockets, Science, and Math

  What child has never wished to play with rockets?  With the move towards using math across the curriculum, rockets are the perfect vessel to create a unit tying math, science, English, and social studies together.

Looking only at the math portion of such a unit, there is so much math we can use in the classroom.  NASA has a wonderful introduction to Rockets covering both math and physics. 

This site has great explanations of functions, the trig involved, maximum altitude, and the Pythagorean theorem.  The index page lists all of the math discussed at this site including area, volume, scalars and vectors, and lots of other math including thermodynamics.

Rocketmime has some simplified equations along with instructions for using them to find how high the rocket will go, time to get that high, and possible mistakes ones can make when running the calculations. In addition, the site provides the equations used in a multistage rocket, creating rocket simulations using spread sheets, parachute calculations, etc.  It is set up to use spread sheets to calculate so much including determining the stability of a rocket. 

If you're not familiar with all the specific terms in Rocket Science, you might want to check out this site from the UK because it tries to bridge the gap and help people understand the physics and maths.  Many of the explanations include examples so you can see how the formulas are used and can take that step towards doing it yourself.

Yes quite a lot of the math is related to science because of force, Newton's laws, etc but this shows students how interconnected the topics are.  The social studies teacher can have students research the history of rockets while they can write up on this in English or perhaps read a book such as October Sky to explore the topic.

Unfortunately, most highs schools are unable to create cross curricular units due to the demands of the state and the federal government.  Even if you can't create a cross curricular unit, you could create a short unit to teach in your math class.

Food Trucks

Food trucks are becoming much more popular than in the past.  If you've ever been to the downtown area of Portland, Oregon you'll see lots filled with them, offering vegan to Thai to Spanish to BBQ.  There are so many you could eat something different for a month.

I've eaten from them when I'm down there for training.  I bet you are wondering where this is leading and how it relates to math.  Well in Volume 53 of Make magazine, there is a fantastic article on building your own food truck from scratch.

The article suggests you decide what type of food you want to serve so you know exactly what equipment you will need.  Serving artisan ice cream is different than BBQ so make a list of everything you need and create a plan for the equipment.  This helps you determine the amount of space needed which tells you the size of truck needed.  You have to know the exact measurements for each and every piece of equipment.


The plan also helps determine where wiring goes, the water, everything so it goes easier.  It doesn't hurt to know any sanitation rules you might have to follow when planning.  Once you have this, you mark the inside of the truck with something like a sharpie so you know where the serving window is, the stove, the sink, everything including outlets. 

The author even recommends hiring an electrician if you don't know much about electricity so its done correctly.  Its hard to change anything once its all put together so do it right the first time.  He goes on to talk about exhaust hoods, gas or propane tanks for cooking, and even finishing touches.

Although there are no prices listed think about the type of project this would make.  You could have students create a plan for a food truck, the costs involved in building it new or used, the cost of the truck, the cost of the electrician.  They might even find out if street food trucks have to pass a health inspection.  Once they've gotten their plan together and a cost sheet for it, they could investigate how much a loan would run to repay it. 

Another real world application of math.  I hope you enjoy the article and you have fun checking it out.

What Math Do Caterers Use?

  The other day, while preparing a warm-up (check Sunday's blog) I wondered what kind of math do caterers use to determine the amount to bill, the amount of food, etc.  So I checked the topic out. 

Where I live, there is no such thing as a caterer.  If you want to throw a party or something similar you either do it yourself or you have a pot luck so I'm not sure if my students even know what one is.

This blog entry makes a wonderful because the author describes the process she went through to cater a friend's wedding.  She had to look at appetizers in terms of the number of bites and included charts on appetizers before a meal, no meal, or after a meal.  Honestly, I'd never thought of thinking in terms of bites. 

How do caterers arrive at the per person cost?  According to this article, it is suggested that once the menu is set and this includes looking at organic vs grass fed vs any other specifics for meat and anything else, the caterer makes a list of all the ingredients necessary to make everything to determine the cost of food.

The general rule of thumb on meat portions is set at 4 to 6 ounces so one has to calculate the total number of pounds needed for the number of planned guests. It is assumed the menu will include appetizers, sides and dessert.   The caterer also has to think about overhead costs such as the cost of electricity, gas, insurance, marketing, helpers, etc. 

They also have to think about any disposable items such as napkins, forks, etc vs the cost of reusable table cloths, etc. One thing that is quite important in the estimate is the per hour cost of the caterer themselves.  Finally is the anticipated profit which must be included.  This gives the caterer a better idea of per person cost or per event cost.

Now lets look at some of these topics in more detail.  For instance pricing can be done either as a per person or per platter which is known as fixed pricing or tiered pricing which focuses on the number of guests planned for.  Tiered pricing is set so the more guests, the less charged up to a certain point.  The mark-up can be done one of two ways.  The first is to calculate the cost of food prep, ingredients, etc and multiply it by three or a straight percent markup of say 25%

We've mentioned overhead costs but one needs to look carefully at the idea of do you include the hiring of extra servers, tables, etc here or do you list them separately as add-on fees so you can keep the base costs down.  Add-on fees might include tent, tables, chairs rental, extra servers, etc. Costs the caterer does not want in the per person charge so as to keep the costs competitive.

One last thing is to consider is if the caterer's prices are higher than others, what is it that makes the extra cost worth it but the bottom line is you can only charge what the general populous is willing to pay.

This article is actually for restaurants but the basics for calculating food costs could easily be used in catering.  It discusses all the things you need to think about from pricing drinks to including drip loss in the preparations and it has a worksheet to use to calculate the cost of preparing a dish.  It has great step by step instructions with reminders of things to consider.

Notice a caterer has to perform quite a lot of math and its a very detailed amount of math.  I think I might take a day in my low performing class and look at this topic so they can see math in action. 

Planning

Warm-up

Real Life Applications of Systems of Inequalities and Inequalities.

In a few weeks, I'll be teaching solving systems of inequalities and one of the first questions a student will ask is "How is this used in real life?"  I always struggle to answer this one because I got my degree in theoretical math so I'm great with the math but not so good with the practical. 

This is used to determine a solution to situations such as figuring out the number of a product that should be produced to create the most profit or determining the correct mix of drugs for a patient.  Imagine its used in medicine.

Its also referred to as the Theorem of Feasible Regions.  In other words, it is the set of coordinate pairs that solve a systems of inequalities.  Its the region which satisfies restrictions placed in linear programming.

So then what does an inequality represent in real life.  In real life, an inequality is recognized by the use of limits such as a speed limit of 75 mph, a minimum payment on your credit card, limit of text messages, or time needed to travel.  For any of these examples they are actually inequalities. 

The speed limit example means you are not to travel above 75 mph but you could easily travel below it.  When you receive the credit card bill, they say you can pay $50 but you can pay more.  This is only a minimal suggestion so your payment is $50 or more.  Many people choose a plan that says they cannot send more than 250 messages in a month or often we calculate our trip to or from someplace based on time. If everything is right, it takes me a minimum of 5 minutes to walk to work but if the weather is bad, it could take 10 to 15 minutes.

In addition truckers face inequalities all the time when they cross bridges and have to keep track of their weight.  An example might be a bridge can only handle trucks whose weight is not over 65,000 pounds.  A trucker has to know the weight of his truck and trailer so he knows if he can use the bridge or must plan an alternate route.

Back to driving but not the speed.  Another inequality has to do with obtaining your drivers license because in most states you must be 16 in order to get one.   This would be a x is greater than or equal to situation but you might have to be 18 to get an unrestricted license.  Along these same lines, there are minimal ages for buying liquor or cigarettes.  Both of these are inequalities.

There are also thermostats in the cars that operate on inequalities also with voltage regulators and even Body Mass Index.

Its easy to find single examples of inequalities but not for systems of inequalities unless you look into linear algebra or linear programming.  At least I have a better idea of how to explain the use of systems of inequalities.  I hope you learned some things because I know I did. 

Absolute Value and Real Life.

  Absolute value graphs as a lovely V shape and often is taught as being the distance away from zero.  The sign tells the direction while the number indicates distance but how is this used in real life?  I never thought about it until I did yesterday's topic and bingo, this hit me!

One example of this used in real life comes from the speedometer.  Say you are a delivery person who travels back and forth between two towns that are 15 miles apart.  Mathematically you may end the day with a total of zero due to a complete round trip is + 15  +(- 15) but the number of miles if you added the values together might be 6 trips at 30 miles each trip would be 180 miles.  So the speedometer looks at absolute value regardless of direction.

Another example is when you are traveling a much slower or faster speed than traffic is going.  For instance you are going 55 mph with everyone else and all the sudden this guy whips out of nowhere going 80 mph.  The difference is 30 mph which is an absolute value because you are traveling 30 mph slower or -30 but we don't express the negative, we only use a positive value.

What about currency exchange.  Often time currency exchange places charge a commission on top of the exchange and it doesn't matter whether you gain or lose value on the money, the commission is always going to be an absolute value.

Take this idea a step further.  In the financial community the money involved in transfers is always treated as a positive value even if the number is negative.  So if an institution loses money, the amount is still expressed as a positive value.  We never say it has a -$600,000 balance, rather we might say, its total loss was $600,000 last year.  The word loss indicated its negative but we do not say -$600,000.


In addition, the voltage in your toaster is an absolute value which means your toaster works properly.

So many ways its used in real life without us knowing about it.  I love knowing this so I can point out its use to make it more real to my students.  This connection might make it easier for them to transfer the knowledge.  I'm off to write some problems for class.  Let me know what you think on this topic!

Step Functions and Graphs

Today in Algebra I, I showed a video on reading graphs.  The guy spoke about step graphs and said this one represented the price per minute and total cost so for the first minute it was 5 cents, then 10 for between 1 and 2 minutes, etc.  Hmmmm, this example made sense but are there other situations which are graphed using step graphs?

When I was in school, we just accepted this was a weird graph you studied but we had no idea what it represented or how it was used.  Today, was the first time I heard any example associated with step graphs.

So the step graph is a visual representation of the step function also known as the Greatest Integer Function which is one step closer to understanding its real world use.  This site has a lovely packet showing some great examples.  For instance, think about the postage on a letter or package.  The amount of postage depends on its weight in terms of whole ounces.  If it weights 5.5 ounces, the clerk will charge you for 6 ounces because that is the next integer.

Another application applies to taxi fees because there is the starting charge and a charge for every 5th of a mile.  So 1.25 is charged as 1.4 miles because its over 1.2 but less than 1.4 miles.  There is also the cell phone where you pay as you go because many of those plans are still set up as so much per minute.

Most tax tables either for sales or income tax are step functions.  If you've ever filled out your own tax tables, you know that at the end you look up on a chart so if you make between $40,000 and $50,000, you pay so much.

If you rent something for so much per hour, half hour, or quarter hour, you are charged for the full amount.  An example might be renting the carpet shampooer from the grocery store.  They charge $3.00 per hour, you use it for 3.5 hours and you pay $12.00 because its over 3 hours so they move up to the next whole hour.

There is bowling shoe rental, laser tax, boating and so many other activities in real life that are step functions. This topic could easily lead into a more specific one on floor and ceiling functions where it goes down instead of up. It opens up so wonderful mathematical discussions.

Let me know what you think.

Best Way to Blend a Classroom.

  The last time I attended a talk on blended classrooms, it turned out to be one of those "Oh we use this program", it does this and this and this and its how we do our blended classroom.

After I heard the talk, I wondered exactly what is the definition of blended classroom.  I've heard the term bandied about but I wasn't sure about it.  It is defined as the mix of technology with face to face instruction.  In other words, it combines classroom learning with on-line learning.

It turns out the flipped classroom is one model of blended learning. There are other models but the one thing advised is for teachers to look at the way they are currently doing things and decide what works best for face to face instruction and what might be better for digital content.  These are very good words.

Education Week has a great article on a 5th grade teacher who integrated blended learning into his classroom.  The blended learning allows him to spend less time teaching because students are spending more time on computers and in small groups.

A general lesson might look contain a mini-lesson focusing on a specific skill or standard based on the results from Khan Academy. If a student has not mastered the skill, they participate in the warmup and mini lesson while the students who have mastered the skill move on to another topic. 

At the beginning of each unit, the teacher posts the list of lessons associated with the standards being addressed.  This way students know exactly what will be covered and can monitor their own learning.  At the end of each class, he has students give a shout-out or a pat on the back for something they worked on.

Based on the description, I would say I have a blended but not flipped classroom because I use technology in the form of apps that work on scaffolding skills.  Most students in one class have trouble solving basic one and two step equations, so I have them work on a game that has them practice this skill while I move on.  Students in my Algebra II class have difficulty rewriting standard form equations into point slope form so I have them working on that skill as we move on to solving systems of equations by substitution. 

I sometimes have my College Prep math class bring their headphones so they can work their way through a unit at Khan Academy.  I use this resource a lot when it comes to solving Trigonometric equations and verifying identities.  It is a great resource for improving understanding and skills. 

So Blended does not always mean you have to rely on a paid platform.  It just involves an online component that helps students learn.  Have a nice day.

The Best Way To Flip A Classroom Without Internet Access

  We all assume everyone has internet access with the number of mobile devices out there but that is not necessarily true.  My home internet is down more than its up and even when its up, I can't always watch videos or download anything at home so I know how frustrating it can be.

I forget where I heard or read this but one teacher puts the material or videos onto a thumb drive so they can put it into their computers.  So one could also put the material on a google site for students to download before they leave school, perhaps at the end of class. 

This is reinforced by an article published on ISTE by Todd Nesloney.  These are his suggestions. Use a thumb drive to transfer material to student computers.  If they don't have a computer at school, he burns the information to a rewritable  DVD so students can get more information without needing new DVD's each time.  For students without a DVD player, Todd suggests recommends loading the material on an iPad or other mobile device for the students to check out. 

In several other articles, it is suggested the material be uploaded to the school's website so students can access it with their mobile devices or post to any free site such as You Tube, Vimeo, or other site that is easily accessible place where students can go.

At my school, students are not allowed to access the internet but there is a way to directly share the material with students who have bluetooth. None of the article suggested the material be shared this way.  There are plenty of sites with information available for doing this.  In addition, there is drop box, etc which are ways can download onto a school device and then transfer it to their device.

Another way might be to pare students up, one who has a device with internet and one who does not so they can work on the assignment together.  This way they both watch it and are prepared when they get to class.  In addition, the link could easily be put into a QR code for quicker access.

In addition, Macs are set up so you can share material fairly easily.  I admit, I've never thought of sharing material in any of these ways.  There are times I'd like to have students watch a video prior to arriving in class but I just haven't figured out how to ensure everyone has access.  I think this coming week I'm going to check to see how many students have internet access and start assigning homework with a QR code attached sending them to the video I want them to watch.

If you have another other ideas, please feel free to share with me.  I'd love to hear them.

How Much Lighter?

How Many In A Pound?

Solving Word Problems

I work with students who are classified as English Language Learners.  I am always looking for ways to increase literacy in the language of mathematics so my students are able to explain their thinking.

For word problems, I love using K-F-C-W or Kentucky Fried Chicken Wings for any problems.  K is "What do I know or What did they tell me?"  F stands for Find or "What is it I'm trying to find?"  C is simply "What do I have to consider or think about to answer the question?"  W is the work.

 Before I  started this process, I relied on identifying the key words but that didn't always work especially if the word might mean one of two operations depending on the context.  This process makes them slow down and analyze the problem.  They have to write down the information so they have a starting point.

Then they have to identify what they have to find.  This means they know where they are going with the process.  They have the start and finish but they don't have the method of getting there yet. That comes from the consider.  They have to think about the operation needed, or do they need a couple different formulas.    This is the part where they plan how to do it. 

The final section is to do the actual work.  I learned about this through one of the books I read.  It is a variation of the K-W-L that works for math.  I love it because it requires the students to read the problem, determine the method of solving and finding the answer.

When I first introduced this, I talk about the process is like planning a trip. The know being the starting point, the find the ending point, the consider as the road map between the two places, and the work as the actual trip itself.  They can related to that especially as they often travel by snow machine and need to plan.

I'd love to hear people's thoughts on this method.  Have a great weekend.

Dividing and Real Life

  I've given several rate, time, and distance problems to the students as part of their warm-up and an interesting topic popped up. Although the answer is in decimal form is it really a reasonable answer for the situation.

For instance, if you travel 1200 miles while going 65 mph, it is going to take you 18.4 hours but are you really going to tell people that.  You are more likely to say it took almost 18 1/2 hours. 

Airlines usually record the theoretical time as 5 hours, 20 min rather than 5 1/3 hours while most people would state the flight was just under 5 1/2 machines.  Another situation is when you are trying to calculate the number of cocoa beans per candy bar.  It might say 140 cocoa beans are used in a pound of chocolate but if you ask how many cocoa beans in a 2 oz bar of chocolate, you'd get 17.5 beans. 

Again this is a situation where we are more likely to say just over 17 beans per bar or averages 17.5 beans because people don't like working with decimals.  What about planning the amount of paint, carpet, ceiling tiles, etc for a house.

We seldom buy 3.2 gallons of paint.  Most people buy 3.5 gallons or 4 gallons of paint so you have plenty to complete the job.  Other times you may only need 2.6 boxes of tile but you buy 3 boxes because they don't sell a portion of a box, they sell the whole box.

So I work on teaching students to do the actual math before conducting a discussion on would we use a decimal answer in real life versus what would we actually say.  One of the students actually asked when would you use a decimal answer for time but I couldn't think of one.  Even in races they break the time down into minutes, seconds and tenths of seconds but its not in decimal form.

I finally settled on the idea that by calculating a decimal answer, it gives us a better idea of what we need.  For instance, if we are traveling, we  know if its more or less than a certain speed or time.  If we are painting a room, we make sure we order enough supplies but we don't discuss it in terms of decimals.

I'd love to hear from the readership in regard to their thoughts on this particular topic.  Thank you in advance.

Dividing Fractions

  The other day I was looking for new ways to present the concept of dividing fractions so students understand why they end up multiplying by the reciprocal.  I know as a teacher I have trouble teaching this because it is one of those processes we teach as the way its done.  I found a way but it is not as clear as this method.

I will start out by thinking Fawn Nguyen and her blog Finding Ways for this method.  It is cool and makes a lot of sense.  Look at the division problem 3/4 divided by 2/3.

Fawn uses rectangles as a way of creating the base for the problem so for 3/4 she recommends a 3 by 4 rectangle and the same 3 by 4 rectangle for 2/3.  3/4th of the squares are colored in to represent 3/4th of 1 while 2/3rds of the other rectangle is colored in to represent 2/3rds of 1.

3/4 and 2/3 are colored in.

 Now you count the number of shaded in squares for 2/3 and you discover 8 squares are colored in.  This is the number of squares that make 1 so you count the number of squares in the 3/4ths and find there are 9 squares colored in so you know you have 8 squares or 1 plus 1/8th left over or 1 1/8

 If you do the math you find it is correct because 3/4/2/3 is 3/4 * 3/2 or 9/8 = 1 1/8

This is one of the cooler ways I've seen to illustrate the concept for division.  Most of the sites simply tell people to invert and multiply but they do not provide a conceptual drawing to help explain.  I like this method and I'm going to play around with it to see if it works in a certain form.  I'll report back in a while on it.  Enjoy playing with this idea.

Cool Web Site For Problem Solving and Word Problems.

  I was out researching something on water sprinklers and ran across a site created by Robert Kaplinsky who has created a bunch of lessons for teachers to use.  He works in Southern California and helps to train teachers to teach students problem solving.

He offers lessons for kindergarten to algebra 2.  These lessons cover a variety of topics and he includes everything needed to use them.

One lesson for Geometry is titled "How did they make Ms Pac-man".  The lesson begins with a challenge and questions the teacher asks students about the game.  Then comes the consider this part where Mr. Kaplinsky goes through the actual lesson, including all the youtube videos needed to support the learning along with his answers.  At the end is the teachers files and the list of common core standards the lesson meets.

Examples of some of the lessons include:

1. "How did he get a $103,000 speeding ticket in Finland?"

2. "How did Motel 6 go from $6 to $66?"

3. "How fast was the fastest motorcycle ticket ever?"

4. "How many Royal Flushes will you get?"

5. "Where would the Angry Birds have landed?"

6. "How can we correct the scarecrow?"

The above six are from Algebra I, Geometry, and Algebra II.  The next 6 are from the K to 6 group.

7. "How many soda combos are there on a Coke Freestyle?"

8. "How much money are the coins worth?"

9. "How far apart are the freeway exits?"

10. "Which pizza is a better deal?"

11. "How big is the worlds largest deliverable pizza?"

12. "How much money is that?"

These all look like fun and are on topics the kids might find fun.  Check it out and let me know what you think?

Does Using Manipulatives Really Help Students Transfer Knowledge?

  When I first read about concrete examples, my first thought was manipulatives.  It has always been recommended students use manipulatives to help them learn mathematical concepts but does it really do this?  Do manipulatives actually help students transfer knowledge? 

Well, lets see!  It was discovered that students may not picture the concept the same way the teachers do who already know the concept.  Students may not interpret the action in the way the teacher hopes and they may not carry the meaning of the mathematical idea.  Unfortunately, students are often taught to use manipulatives in a rote manner.

Manipulatives work best if reflect on their actions with manipulatives.  This helps them build meaning.  This is integrated concrete knowledge which combines separate ideas into an interconnected web of knowledge.  In other words, mathematical concepts become integrated concrete by how well they are connected to other ideas and situations.

It does not matter whether the manipulatives are virtual or real, the students who use them need to build the ideas from the manipulation of the representations and thinking about their actions. 

The one idea that comes up again and again in literature is that giving the students the manipulatives does not guarantee they will understand the concept.  It requires careful planning and good transitions.  The teacher needs to remember the purpose of the lesson when using manipulatives and must create a carefully thought out sequence of steps.

It was found that the higher the level of guidance the teacher provides, the better the learning outcomes in retention and in problem solving.  When they reach a point where they require a lower level of guidance, their are better able to transfer their knowledge.

So all in all, if we use manipulatives in math, we need to make sure that we carefully plan their use so the students gain the knowledge we want.  If we do not, then we may just be using them in a rote manner so students may not learn.  I'd love to hear about your experiences.  

How Many Gumdrops

How Many In A Pound?

Blood Splatter Trigonometry.

  The other night, there was a shooting and suicide in town.  Since this is such a small place, most everyone is related to the victim and the person who took his own life.  To make things easier for students, I showed NUMB3RS.  The episode itself was on Magic but Charlie mentioned Blood Splatter Trigonometry.  Have you heard about that topic?  I hadn't until this moment.

Blood Splatter Trigonometry uses the properties of math to determine a lot of information from just the splatter.  The size of the stain tells you about the type of energy involved.  For instance, a small stain indicates a high energy transfer while a misty looking stain could indicate an explosion or gunshot.

In addition, the shape of the splatter indicates the angle from which the blood came.  For instance, if it dropped straight down and hits at a 90 degree angle, the splatter will be circular while an elliptical indicates less than 90 degrees.  Further more, trigonometry is used to find the area of origin or location of the blood source and to find the point of convergence or where two or more blood stain paths intersect.

Cornell has a great lesson on this topic which focuses on the NUMB3RS episode and it includes three activities along with great explanations and the math equations used.

This Prezi shows the actual math involved in using the blood spatter trigonometry.  According to the presentation they can determine the origin of the blood, the weapon, the extent of the attack, and the height and dominant hand of the attacker.  There is a great chart showing what the blood looks like if its dropped from various angles and the angle involved in creating the right angle triangle to calculate things.

The scenario is set up  and the reader is taken through the process step by step to see how it is done and all the equations are shown.  It is well done and is perfect for including into a class if you want to show your students how trig is used in real life.

Finally, I found this 9 page paper which goes into this topic in more detail with wonderful examples so a teacher could easily include this during a unit on trig.  This paper goes into more detail on low, medium, and high impact, angle of impact, area of convergence and determining height.  I found the information in the paper is quite interesting.

Since I have a class doing trig, I think I'm going to schedule three or four days to cover this topic in detail so my students get to experience a real life application of trig ratios.  Check it out and enjoy.

Smart Exchange.

  I am a proud owner of a Smart Board attached to my wall.  I use it for so many things from showing videos in class, to creating units with notes to playing jeopardy.   I don't think I go a day without using it.

For those of you who don't know, the Smart Board people offer a place where you can post materials or download materials.  Its called the Smart Board Exchange and is filled with materials for almost every subject.  A few presentations cost money but for the most part, they are free.

You can type in a topic or you can use an advanced search which allows you to narrow your search to topic and grade.  I typed in trigonometric graphs because we'll be doing that in the advanced math class next week.  I came up with several possibilities such as a 12 slide presentation on graphs to a 55 page one that shows the relationship between the unit circle and the graph of the trig ratios.

The site is set up so you can preview the material before you download it.  Sometimes the material I want is spread out over two or three presentations, so I integrate the slides I want into one presentation. 

Many of these presentations are interactive, have movement, vocabulary, questions, both guided and independent practice so its not just you show and they watch.  Students often get up and get to interact with the board.  I've actually written clarifying information on slides when my students had trouble understanding the material. 

Since I work with ELL students, I sometimes download a grade 5 presentation because it provides the material at the right level for their understanding.  Other times, I might download the lower grade because it is filled with animation that is perfect for getting student attention.

If you have a smart board, check the site out and find all sorts of great material.  It might save time as you prepare for your weekly lessons. 

Can We Teach Students Not To Multitask?

  We are all aware of the myth swirling around society.  The one stating we can all multitask but those of us who stay abreast of research know it is a myth.  Although you may being trying to do a variety of things at once such as surfing the internet, texting on your phone, listening to music, or reading a book, we know you can only focus on one thing at a time.

Recent neuroscience research shows that our brain is not capable of multitasking in the ways we thought possible.  Even attempting to do two things at once is still not possible.  What we think of as multitasking is not.  What actually happens is that every time we change our focus, our brains go through a start and stop process that in the long run wastes time.  We actually become less efficient. In addition, we make more mistakes and it can sap our energy.

So how do we convince our students not to multitask.  My school does not allow the use of mobile devices in school during the day except for lunch because students want to spend time checking for texts or listening to music.  In one study it was discovered students could not go more than two minutes before checking their facebook account or reading texts and by the end of 15 minutes only 65% of their time had been applied to studying. 

This same article indicates that results from various studies indicate student learning is not as deep or good as if they apply their full attention to studying.  In addition, they understand and remember less and are unable to transfer knowledge as well as they should. 

My students are always telling me they need their device so they can listen to music while they study.  I allow it in the evenings during study hall and I've noticed they spend more time actually switching between songs than they do studying.  When I point it out, they tell me it is not interfering with their studying.

So I struggle with convincing my students they are wrong.  I found a lovely activity at Psychology Today to show them multitasking is a myth.  It is easy to do and you might change it a bit to let them text while doing it.  I don't know if I'll convince them of it but I'll keep trying.

Common Core Testing Dropouts

  According to Education Week, the number of states participating in the standardized PARCC and Smarter Balance Tests are dropping.  I know our state had signed up for Smarter Balance and then changed to a different test that had its own issues and we never implemented the test.  Last I heard, the state of Alaska is working on something else.

So why are state dropping out of Common Core after so much time was spent implementing it?  Some of it has to do with dropping scores.  Since the new testing focused on different standards and on different aspects, students were not as prepared for these tests so the scores showed a decline.  Unfortunately dropping scores are often seen as the failure of a district to teach students, not as a possiblee adjustment  to a new type of test. 

People forget that anytime a new curriculum or a new testing system is implemented, test results decline.  It can take 5 to 7 years for any new program to start showing real results because everyone has had a chance to use the program properly.  We cannot change tests and expect the same results as before. 

I checked out some practice questions from the Smarter Balance test and I knew my students would not do well because the questions were expressed in ways they were not used to.  Furthermore, there were questions using methods I'd never seen before and it took a while to understand what was being asked.  If teachers are having difficulty with the way questions are posed, just think about the students.

Since there is a push to perform well on any standardized test, some states have changed the rubric for  interpreting the scores so results are more favorable looking to the public.  Although this defeats the purpose of a company created rubric, one can understand a states desire not to appear to be a failure in educating students.

Furthermore, most of these tests are switching from pencil and paper based to computer based which is great if you have an internet system that is guaranteed to have the necessary band width, enough computers, and the provider does not run into issues.  Last year, the computer based tests were cancelled at the last minute because the provider ended up having a line cut and we couldn't access them.

So where do we go from now?  We need to choose a course of action and stick to it rather than try this or that. What is your opinion on this topic?

Labor Day Warm-up

House Flies

Cocoa Beans

Concrete Examples in Math

  Using concrete examples is the last of the six learning strategies but it is a topic that sounds easy at first but isn't that easy to define a concrete example in mathematics.

Often times the concrete examples are explained using the application or effect it has in a situation.  The Learning Scientists in their blog use the example of scarcity to demonstrate  its meaning using airlines as their concrete example.

So in Mathematics, concrete examples often manifest themselves as real world examples such as the use of the parabolic flight path in Angry Birds.  You have to choose the correct path so you wipe out the enemy.  This is really easy to find concrete examples.  When studying ellipses, you only have to look at planetary orbits but finding a concrete example for hyperbolas are a bit more difficult.  However, circles are fairly easy by talking about certain types of buildings.

Ratio and proportions are all over the place from reading scale models and finding how far a place is based on a map. Percentages, mark-ups, discounts, are found all over the place in the newspapers, on the television, and so many other places.  In addition, finding applications for area and volume are just as easy but what about some of the other topics like multiplying binomials.

One concrete idea I have is based on the idea of planning a square room and determining the change in the area by changing the measurements say adding three feet to one side and subtracting 2 feet from the other so you have a rectangular shaped room.  Perhaps you are planning to build a house on a narrow lot and you need the house to be so many feet from the sides and back.  I think you could use binomials to help figure out the area of the house.  This is a much harder topic to find concrete examples for.

On the other hand, trig is seen in surveying, flight, and so many other real world applications that you really don't have to search.  When I look for concrete examples, I try for ones that make sense to my students and seem real rather than made up.  I always hated those train problems.  You know the ones where one train leaves at a certain time and a second train leaves two hours later from the opposite direction.  They never made any sense to me.  If someone had explained I needed to know that so they could have the switch ready so the trains didn't crash, I would have been happy but the problem itself made no sense to me without the context of why.

Think about it.  Most railway lines are only made for trains to go one at a time, so if you have two trains using the same track, you have to figure out where to have them pass safely.  Linear equations are nice because business provides us with activities.  Unfortunately some of our older examples such as rental cars are falling into the past.  It used to be they would charge a daily rate plus an unlimited mileage rate but now its usually a flat rate with unlimited miles.

That is the only thing with concrete examples.  My concrete examples often change due to companies changing the way they do things.  It makes life interesting.

I am still deciding if manipulatives provide concrete examples or if they provide concrete visualization for students.  I'm off for the day, have a great weekend.