Real Halloween Math

As a high school math teacher, when I look for Halloween math, I am not looking for those problems that are just regular problems with pumpkins.  I want something more real.  Something like "You grew a pumpkin that weighed in at 1810 lbs and someone offered to buy it for $0.23 per lb.  How much money is it worth?"

In that vein, I discovered a lovely article on Smarties, a candy we all grew up with.  The level of this material is actually geared for elementary students but if you work with students who are ELL or who are below level, this is great.  It begins with a link to a You tube video which takes everyone on a tour of the manufacturing facility.  There is also a nice 2 page article on the manufacturing process.  The level of reading is perfect for my lower level Algebra I students.  It concludes with a nice worksheet on Powers of 10 which is an area that some of my students are weak in.

Yummy Math has a lovely worksheet designed to have students calculate how long it will take them to cover the whole neighborhood.  It is a very realistic task for most students but in small Alaskan villages, we don't always have streets or neighborhoods like these.  I might adjust this one using a small map of the village itself so as to make it  easier for my students to relate to.

This site also has an activity for students to calculate the volume of a variety of bags and other containers to see which one is best to use to collect candy.  That is one my students will go for and love doing it.  There is another that has students look at a variety of packages of candy.  They need to decide which is the best deal and the worst deal.  Comparison shopping which is something my students do not know how to do because of the having only one store in town.

There are several other activities which are awesome including several on things like if a restaurant is going to raise a million dollars on Halloween, can they actually do it based on certain things or how many pies will a giant pumpkin make, or interpreting graphs on Halloween candy sold.  Real life examples of math associated with Halloween.  Yes!

I know what I'm doing on Wednesday when I have my world problem day.  Have a good day and Happy Halloween.

Warm-up

Warm-up

Reading and Math

  I don't know about you but I've had to teach reading in my math classroom because I work with ELL students.  In addition, I like to show students the forms that math appears in the real world.  Well, yesterday another teacher and I provided a 3 hours workshop at school on differentiation.

We covered the usual material such as tiered assignments, choice and menu boards but we provided another resource that most people are either unaware of or they don't know how to use it in their classroom.

The site, NEWSELA, is filled with tons of news reports from all over the world.  When I checked it out to get the URL for this blog, the first article I saw was on using drones to get aid where needed.  If you read this blog, you know I love technology but where this site differs from others is in that you can print each article out at a different reading level or place the material in a google doc for your students to download.

I typed "Mathematics" into the search bar and found a great article connecting art, math, and science.  It is offered at four different reading levels from low to high.  In addition, it offers a writing prompt designed to go with the reading level of the material.  It also offers a quiz to go with the material.

For my students I would create a worksheet of questions they can fill out as they read the material otherwise many will just skip through the reading, fudge the answer to the prompt and then whiz through the quiz.

I tried statistics in the search bar and discovered an article on the statistics of baseball.  It looks at how the score board displays information on the game and how it could be interpreted.  They author even went into how things have changed over time.

Then there is a second article focusing on the different forces in play in a baseball game.  Although the results do not always apply directly, there are so many articles you are sure to find some you can use in your classes.

I am going to start using these in my math classes to help improve their reading abilities and show them math is found outside the classroom.

Give the site a check.  If you like what you see, sign up for the free version and enjoy.

Grids and Coordinate Planes

The other day I watched one of those police shows where they created a grid of people to walk through an area, hoping to find the body of a missing person.  This reminded me that grids are used in real life.

We see grids used all the time but we don't always think about its uses. Although this is not true of my local phone book, many phones books for larger cities include pages of maps set up with a grid overlay.  Usually one axis is labeled with letters while the other uses numbers so the store you are looking for might be in section A-3.

Archeologists and geologists tend to divide the land up into grids so when they find something or make an observation, they can make note of it through a coordinate system.  Landscapers plan using a coordinate system because they have to know where the house, garage, and other buildings are before deciding the location of all the plants and trees.

Even digital photographs and screens are created using pixels which are another way of saying coordinates for each unit.  You hear the resolution might be 1290 by 780 pixels which means you have an x and y location for every single pixel in the picture.  The same idea works for video games and where everything is located.

Back to mapping.  All maps tend to have some sort of grid associated with them in the form of longitude and latitude.  For instance there are certain parts of Utah, especially places around Salt Lake City, use a grid system established in the mid 1800's when the temple in Salt Lake City was designated as the origin.  Most unincorporated places around the city assign addresses based on this grid system which uses North, South, East, or West instead of the X and Y axis.

In addition, there is the military grid reference system or MGRS, a geocordinate system used by NATO to locate places on earth.  The location is given using an alphanumeric string.  This system is derived from the Universal Transverse Mercator coordinate system.  It is said the United States  National Grid system is easier to read since the coordinates are written with spaces so the coordinates are easier to read.

Don't forget that search and rescue often divides their searches up into regions or grids to improve the chances of finding people.  With the availability of GPS, it is becoming more and more common to use that to help establish the search territories.

So much and I've only touched on a few areas grids are used in.  Maybe I'll be back to explore this topic.

Measurements

  Did you realize that looking at measurement is the perfect time to introduce a bit of a history lesson so there is a cross curricular connection?

Just think about where the first measurements came from?  One of the earliest is the Cubit which is the length from the tip of the finger to the elbow.  Hmmm, everyone could measure their bodies to find out how much the cubit varies within the classroom.  It wouldn't take much to start a discussion on what problems these differences might make.

A cubit could be subdivided into the foot or hand.  The hand is set at 4 inches and is the standard measurement for the height of a horse so if you horse is 15 hands high, it is actually 60 inches tall or 5 feet.  A nice exercise built to create fluency for converting between systems of measurement.

The yard has existed for a long time but it was standardized during the reign of Edward I when it was determined that 3 feet = 1 yard or Ulna.  In addition, the foot was declared to be one inch but the inch was based on the length of 3 barley dry barley corns. 

It was at this time, other measures were standardized such as five and a half Ulna equal a perch also known as a rod.  An acre was declared to be 40 perch by 4 perch.  Wow, think of the fun one could have figuring out the number of feet in that acre and comparing it to the current definition of an acre!

Interesting fact - the perch was originally defined as the total length of the left foot of the first 16 men leaving church on Sunday.  That would make a cool exercise in class to see how it would work with modern man.

Another interesting fact - many of the standard measures for meters and yards were made of metal but depending on the type of metal, the official standard could possibly shrink by 1 part per million every 20 years. 

Beginning in the 1500's or so, countries began working on standardizing all measurements so there was a consistency of use.  France was one of the first countries to work on standardizing measures because there were over 250,000 different units of measurement being used.  That could be so confusing.  They defined the meter as one ten-millionth of the distance from the north pole to the equator. 

Eventually after several years, the official length was made and the rods to represent the official meter were made out of plutonium.  The United States claims a foot is 0.3048 of a meter which sort of makes sense since a meter is about 3 1/4 feet.

I love the idea of asking my students "Why is it important to have standardized measurements?" I can hardly wait to see their answers.  I'll let you know some of the responses.

I'd love to hear what you think.

More Ways to Develop Number Sense

  Since I teach High School, I decided to look for additional ways to help high school students increase their number sense.  Students who are low performing or perhaps English Language Learners need scaffolding so they develop a better number sense.

While looking for additional information, I discovered this 17 page pdf with some really good suggestions including the idea of what most teachers mean when they say "show me your work."

In a sense, I am guilty of what the author charges but at the same time, its hard to break with traditional ways of teaching the way I learned things.  I do like some of the activities such as the one on representation which shows four different ways of approaching the same problem.  This appears to be a sample of a much longer book but its enough to help the teacher approach the topic in a different way.

Here is another article from Yale on ways to improve number sense in the high school.  The author discusses in detail  various topics related to number sense and includes a wide set of practice problems.  Each set of practice problems focuses on a specific skill.  I have high school students who are unable to subtract if they have to borrow.  I am going to teach students to count back rather than try to borrow because it is actually finding the difference.

I looked at a presentation on improving number sense in the middle school and one line stood  out above all the rest.  "Always put number in context"  Instead of saying "What is 20 \ 1.79?"  Ask it as a word problem such as "How many composition books priced at $1.79 can you buy with $20?".  This puts the problem in a more realistic setting while working on helping students learn to work with word problems.  Furthermore, many of the new tests are more likely to use the problem as it appears in the second example rather than the first.

The same site discusses ways to talk about numbers.  Start with smaller problems to get students thinking from a variety of perspectives.  Discuss a strategy offered by another student.  Limit talks to no more than two minutes and remember it is fine to put a strategy on the back burner till another time. 

According to a different website, you can tell how well as freshman student has done previously in math by asking him or her if they can estimate the number of objects in a group.  Such as simple test, yet I didn't know about it.  I often have no idea what my incoming students know and this will help. 

So enjoy and let me know what you think?

Is Rate Of Change Really The Same As Slope?

  Over the past week or so, I've introduced my Algebra I class to slope and interpreting slope in real life situations.  This actually lead to a question from one of my students. He asked if slope and rate of change are the same?  The two are used interchangeably but are they really the same?

I told them I see rate of change as something that is more real world such as miles per hour, or typing per minute, or any rate whose change is constant while slope is the mathematical representation of the real world rate of change.


I answered it the way I did because we usually teach the topic with slope always being associated with the line on a graph but if you think about it, most slope in real life is already calculated such as the pitch of a roof, or the grade of a mountain road. 

In the past, I've usually just taught the slope mathematically with a visual graph but I've not extended it into the real world.  I've thrown in a few graphs for students to interpret but nothing more.  This year, I've teaching slope hand in hand with real rates of change and with interpretation so students get a better background in the topic.

This week is short week, so I've planned a few activities such as
1.  Calculating slope from a topographic map because the rate of change determines the grade for a road, the rate of erosion and other things.  It helps students see that rise over run in this case has real meaning.  The rise is a change in elevation, while the run is a change in distance.  This site has a nice explanation, instructions, and practice problems.  It covers math for the student who is studying the geosciences.

2. Calculating the slope of a roof after its built.  WikiHow has three ways to do this and emphasizes the idea that rise and run are both distances. The information has good pictures and instructions.

3. If you've ever watched the news, they always report on the NASDEQ or Dow Jones, both of which has slope but how is that calculated?  It turns out, the lines are actually a line of best fit using regression as explained here.  To simplify this, why not get the daily prices for one or two stocks such as Coca-cola or Apple to see how they move up or down and calculate the rate of change for those?

4.  Prices of gas go up and down in a cyclic manner.  Monitor local prices and calculate the slope of the increase or decrease in price over say 30 days of even a year.  Where I live, the price stays quite stable and only changes once maybe twice a year when the barge arrives with the fuel or the store put it on sale to get rid of it before the new barge load arrives.

5. Of course there is always having students calculate the slope of the ramp at school or any stairs as they are both rates of change.  These two are easy to find and use without much trouble.  You could even have students look up the law to see if the school's ramp meets ADA requirements. We have a ramp but there is a nasty turn in it so I'm not sure if it is fully legal.

So over the week, my students are going to measure the ramp, the stairs, and do a change in elevation off a topo map.  I'll let you know how it goes.  Let me know if you have any other suggestions for students to experience rate of change.

How Much?

What is my volume?

The Importance of Number Sense

  I've noticed that each year, the incoming freshman class has little or no number sense.  They do the math and whatever the answer is, it is right in their minds.  I am not bemoaning the job their previous teachers did because of the books they had.  They've gotten a whole new program which will work on that but it will be a few years before results start showing but in the mean time I need to work on that now.

Number sense is defined as being fluid and flexible with numbers.  It is a skill needed by all students to do well in math because with a strong foundation in number sense, students do better.  The ones who struggle with it, have never really developed it.

With an undeveloped number sense, students are unable to develop a strong foundation even in mathematics let alone higher maths.   These are the students who struggle when they get to high school, potentially choosing to drop out.

Why be concerned with it?  It is something needed in real life to ensure success at what ever job is chosen.  With out it, it leaves adults unqualified for many of today's jobs.  So what are ways that we can use in our classrooms to help develop number sense.

1.  Model different ways to do the computation - This exposes students to ways to solve problems they may not have thought of.  One of the methods just might be right one which allows the student to "get it".

2. Encourage mental math because it helps students develop the understanding of numbers and their relationships. At the same time, have the students discuss how they found the answer in their head.  Take time to talk about the numerical relationships used.

3.  Lead classroom discussions on strategies for solving equations because it helps crystallize their own thinking and consider classmate's approaches.  Be sure to write ideas on the board so students see the connection between mathematical thinking and symbolic representation.

4. Encourage more estimation in school because it is used all the time in real life.  Note that estimation is not always about rounding. Encourage them to estimate an answer for calculations before completing the math.  My students love doing the math before rounding to create their estimate.  I'm working on turning that around.

5.  Ask students to share their reasoning both for mistakes and for correct results because math needs to make sense to them.

6.  Pose questions that have more than one right answer so they can see that not every problem has a single correct answer.

It is important to encourage the development of number sense.  Although it should happen in elementary, it does not always work out that way.  Integrate it in your classroom to encourage those whose number sense is not as advanced to improve and to fine tune those with a decently developed number sense.

What do you think?

Drones in the Math Classroom

  I own my own mini drone.  I haven't gotten to play with it as I need a nice open spot and I keep forgetting to take it out to the school gym when no one else is there.  So are there ways we can use it in a math class and enjoy making the math fun.  That is a big yes!

First of all, the owner of Ziplines just started a drone delivery service in Rwanda to get blood, plasma, and anticoagulants to remote hospitals. The cost is the same as if it were delivered by land but the time is cut down tremendously using this method of transport.  In addition, the costs are even less because the drone does not land but drops the material via parachute.

So what are some ways you could use a drone in your classroom.

Students can make their own version of the powers of 10 video where they start at ground level and then move out 10 times a distance.  The drones would allow them do do quite a few outward movements.

Drones could be used to to help illustrate the speed or rate times Time equals distance by conducting a series of experiments where students create a straight line of known distance and record the time it takes the drone to go that distance.  They can find the speed of different situations.

Another is to record distance of a path on a coordinate plane to find out if the hypotenuse is shorter than the two legs.  This is a way to show a practical application of Pythagorean theorem.

Check out this video on the Teaching Channel which talks about using a quadcopter as a way of showing sin, cosine, and the laws of sines and cosines.  It is a great video covering this material in a way I've never seen.  They look at the sine and cosine in terms of what happens to the rotors as it goes up, down, left, and right.

It discusses three different scenarios which use the drones placed in specific real life situations.   These scenarios require the use of law of sines and cosines in order to solve the problem.  It is great and specific enough anyone could use this type of project in their classes.

Students could use a drone with a camera to survey school property.  So students would need to know how the information from a drone is used so they could repeat the process.  Math is definitely used in surveying.

These are just a few ideas and I'm sure there are quite a few more out there. Now you have an excuse to use a drone in your classroom.  Go out and have fun.

Promoting Self-Correction

Now that we know we need to explicitly teach students to identify and correct errors, the next step is to help students create a self-correcting mindset.  

A good introduction to the topic can be found in this video at the Teaching Channel  The two instructors explain how they use questioning to promote self-correction and to help their students become more independent learners.

One way to help students gain the ability to self correct is with a slight change to our thinking as teachers.  Allow students to work a set number of problems together if they want, or let them work independently.  Its up to them.  When they've completed the assignment, let them correct their assignment using the teacher's manual.

Students use markers to correct their work so one color indicates correct, another for incorrect.  Allow them to write in the correct answer.  Let them journal about their experience and have them include thoughts on their thinking process and where and why the mistake happened.

Another suggestion reflects back to yesterday's entry on having the students create a check list so they can go through each problem checking for the places they are most likely to do the math incorrectly.  The checklist gives them a place to start.  This is something I need to do with some of my math classes.

We know that when students find their own errors, they become more empowered.  Unfortunately, finding errors in math is not always that easy as it can be as simple as missing a negative sign or as complex as not substituting the correct variable.   I noticed that many of the websites I visited when researching self-correction focused more on practicing certain skills rather than focusing on teaching the skills to think about why the mistake was made.

Even teaching students to self correct means we have to change the attitude of wrong is failure and its ok to go back and figure out why students don't get the right answer.  Out here, students play lots of basketball.  I just have to show them that self correcting in Math is the same process as the one they use in basketball. 

If they miss the basket, they stop to ask why they missed it.  They think of possibilities and once they've determined them, they begin practicing so they can get better.  I sometimes think I need to teach math as if its a sport and they are trying  to go to state.

Let me know what you think.  Thanks for reading.

Changing Perceptions of Mistakes Part 2.

   Yesterday, I discussed societal perceptions of math being only right or wrong so if you make a mistake you are wrong and a failure but as we've seen that is not always true.  Making a mistake is just a signal letting the individual know it may be a misunderstanding of the math process, it might be a sign that was lost, or numbers switched.

Today, I'm looking at ways to help students learn to identify the type of mistake they are making so they can work on becoming more proficient.  It is not easy to work something else into your day but I figured out how I can use some of these suggestions with my students.

1. Write a problem on the board with several solutions.  Ask students to rank the solutions from best to worst. Next have the student discuss the ranking with their neighbor and create the criteria for ranking the solutions.  Finally, create a list of the most common mistakes and suggest ways to catch them or prevent them.

2.  Assign a select number of problems for students to complete.  Ask them why they think their answer is correct or incorrect.  Ask students questions to help them see different ways of reflecting on their thinking. 

3.  The teacher needs  to monitor student work to see areas of misconceptions so the teacher can help clarify those areas.

4.  Take time in class to create a list of the top ten mistakes they do such as multiply the number by two (the exponent) rather than squaring it.

5.  Help individual students create a list of their own common mistakes to use as they do their work such as switching digits in a subtraction problem so they don't have to borrow.

6.  Have students mark the spot where they run into problems as they work the problem so if their answer is incorrect, they can return to see if that is where the mistake occurred.  This also identifies something they can ask for additional help in learning because it lets them know what they still don't know.

My final thought on this is that I have not explicitly taught my students to learn to analyze their errors.  I have just started including one math problem in their warm-ups that will have an incorrect answer.  Their job will be to determine where the error occurs and what the exact error is.  How can I require them to find their own errors if I have not taught them the process?

Tomorrow, check out teaching students to self-correct.  The first step is to change people's mindset from getting a problem wrong is failure to an incorrect answer is actually just telling us where we need a bit more work.  Teaching them to self-correct is the next step.





 

Changing Perceptions of Mistakes, Part 1.

  Does this sound familiar?  You teach the material, give work, correct it and hand it back.  The work goes into a folder or possibly the trash but you don't have any sort of follow through requiring them to correct their mistakes.

The other day, I realized the only time I require students to perform an analysis of their mistakes is when they want to retake a test.  Otherwise I don't do it. Its mostly because I literally don't have the time.

To effectively learn the material, it is not a question of how much practice, its more a question of the type of practice done.  In other words, its best to determine what is not working and mastering those before moving on.

In school and in math, most students react to mistakes emotionally and often feel stupid.  This means we need to look at removing the shame associated with mistakes so students will look at the mistakes to see why it was made rather than hiding.

It is important to get students to look at the errors instead of stating they bombed the test.  In math the mistake is something as simple as dropping a sign.   One way to help students look past their shame is to take one or two of the most common errors made by the students and analyze them as a whole class.  Look at the specific mistake together.

It is said that mistakes have a concrete basis such as ignoring a sign, forgetting your multiplication facts, even forgetting to borrow properly.  Teach the students the teachers mark is simply a sign saying hey look at this and see where you made the mistake in the process.

Many teachers do not require that students check their work.  Until I read this, I have often allowed them to do the check in their heads but after reading this, I realized its easy to make a mistake in your head.  Starting this week, all students will be required to check their work when appropriate.  This is not a habit I used in college or in most classes I've taken.

Its interesting that even know for all the discussion we have that students need to learn the concepts and the full process, many of us still promote answers as either right or wrong.  When I was enrolled in teacher training classes back in college, there was a huge discussion on do you take the stand of  the answer is completely right or wrong?

Other training I took along the way enforced the same idea for tests but with the added idea of having students go through the test and figure out where the mistake was, rewrite the problem so its done correctly and explain where the mistake occurred.  It is only as I write this entry that I realized no one has ever talked about teaching students to find their mistakes.

I think that may be the missing key in helping students see that making a mistake is not failure but a step towards seeing it as an indicator of focus to learn that part better.  Tomorrow I"m going to discuss ways to teach students to check their work for weaknesses rather than mistakes that caused the calculation to be wrong.

Volume of Bubbles.

How Far Is It?

General Thoughts on Technology.

  I read somewhere that the human mind is no longer remembering as much as it used to because mobile devices are being used as our memories. 

Someone pointed out today's devices have internet so we can just look it up if we forget something.  We have calendars with reminders that pop up to keep us on task.  There are apps that scour the internet for very specific material so we don't have to do that anymore.

So what effect does that have on mathematics and learning?  It has some.  In many places, students are no longer required to memorize their multiplication or division facts since they can just pop the problem onto a calculator that does the math.  In fact, there are quite a few calculators for all sorts of math problems you can use to have them solve it without you ever really knowing the process.

I have the My Script calculator which will solve a variety of problems from simple one step equations to the law of sines and cosines.  They just type in the problem and place a set of parenthesis for the variable and it solves it.  Most of my students do take time to learn the process but a few rely on the calculator's answer.

Unfortunately, this means they do not develop the number sense needed to know if their answer is even in the ball park.  They accept the calculators' answer as the truth. So I'm still working on creating a balance between learning the process and developing number sense with using calculators and asking yourself "Does the answer make sense?"

I've found some great apps which do reinforce skills but my students do not like the ones that provide an explanation of why they missed the problem.  They would rather just move on and do the next activity without trying to correct their mistakes.  I've thought of providing them with a sheet to write down all work and if the answer is incorrect they have to explain where the mistake was made.  The last step would be to write out the correct way of doing the problem. 

This might help them slow down and reflect on the type of mistake they are making.  I don't think I have students do enough error analysis.  At a training I had years ago, they said it was important for students to learn to do the math correctly. 

I'd love to hear your thought on these topics.  Do you think that deeper understanding is being slowed down by easy access to tools which solve problems for us?

Proportions and Real Life

  We teach students ways to determine if two ratios are proportional but do we ever really spend time showing students its multiple uses in real life?

Out in the bush, the first application of proportions is in making clothing.  They base it on a person's own body unit such as inch for the finger, or yard via the arms.  They say they create the pattern this way so its right for the wearer.

Another situation is when you have to enlarge or decrease a recipe.  You use proportions to determine the new amount of each ingredient so your yield is the right amount.  Often, I find recipes created to make food for a family of 6 and there is only me, so I reduce it by about 2/3 so I've got enough for me for a couple of meals.

At the end of a snowstorm, its easy to calculate the amount that fell on average each hour, during the storm.  Again, its the total amount/total time = amount/hour which is often done by the weather channel.  Its much easier to do an average rather than trying to calculate it hour by hour.  I don't know if you've been out in a blinding snow storm but I have and I never stop to figure out how fast its falling.

This particular example comes up in a variety of math but its true that currency exchange uses proportions.  You have the exchange rate of say $1.00 American = $.70 Canadian so how much would you get if you wanted to change $35.00 American into Canadian?  I know about those exchange rates because my sister used to work for a customs house and I had to help her figure out how much to bill a customer in Canada.

Proportions play a large part in art when an artist uses proportion to get the look right on items in a picture.  In Ancient Egypt, they proportioned the body so it was in thirds, even the children so the body from waist up was 1/3rd, the waist to the knees, and the knees down.

Even architects use proportions to create buildings that "look right"  If the proportions are wrong, the building looks wrong. If you check out this place, it is a download which looks at the different types of proportion and ratios used in Architecture.  It comes with worksheets and explains the application of the golden ratio as used by architects.

Its nice to have items to discuss in further detail when noting real life applications of proportion other than the standard ones from the textbook.  If you find more examples, let me know as I"m always open for suggestions.

Math Game Apps - How do you find the right one?

  I think we all struggle with this question.  I've read tons of recommendations via lists from this authority and that authority but too many were elementary or cost more money than my school is willing to pay or I don't want to get it because it costs and I like trying it out before buying it for my ipads.

There are often light versions of apps which allow you to try them out before buying but they often only allow access to the first level or two.  You have to pay to check out the locked levels.

I think those lists are a good place to start but you need to know your students.  I've made mistakes.  I put a multiplication practice app on the iPads and realized later  that you just had to type in the x 1, x2, etc in order.  The app did not work as I thought.

Before I look for an app, I ask what skill am I wanting them to learn or  is this app going to be used to scaffold students?  Is it going to be used more to introduce materials or be used for ongoing practice.

Once I've determined the need the app is to meet, I look at the following.
1.  Does the description tell if it will meet the need that's been determined?

2.  Does it come with instructions so either I or the students can use the app?  I've had apps I couldn't figure out at all because the instructions were poorly written or they were nonexistant.

3. Does it have good reviews?

4.  Is the cost reasonable?  If its a lite version, does it give you a good idea of the locked layers of the app. 

5.  If you plan on using the lite app does it let you use those skills or does it require you to buy the app at a later date?

6.  Is the level such that my students will be interested in using it or is geared for too young a student?

7.  Does the app require students to register for their webpage in order to use or does it need to be online to talk to the website?

I've downloaded a bunch of apps that sounded really good but when I tried them, they were not as described.  It is important to check the apps out before putting them on the iPads.  I tried to find articles on the criteria a teacher should use when selecting apps for the classroom.

To summarize, select the apps which will meet your requirements for your class because you know your students.

The Math That Got Us To Mars.

   Did you see that movie "The Martian"?  It was not too bad but since the movie came out, there has been quite a lot of information provided on the planet but what math is required to get us there?

If you want to build interest in doing some of the math, start with this wonderful article which looks at the real life science of the craft and the mission in the movie.

 If you want to show how important precision is when planning a mission, because a small error can cost a huge amount of money.  According to an article in the LA Times some engineers forgot to convert standard measurements to metric and the $125 million Mars Climate Orbiter was lost.

The JPL navigation team did their calculations in metric while the Lockheed Martin chose to do their calculations in standard English measurements such as inches and feet.

Unfortunately, the error caused the craft to miss Mars.  Just a small error can have far reaching results.

This 10 page pdf has some wonderful information on orbits, planning to get a spacecraft from the earth to mars along with the mathematical equations used for the trip.  It introduces Kepler's laws which deal with the planetary travel, Newton's laws of motion, and includes lots of illustrations to help people see what is happening.

Another good site is the Rocket and Space Technology page on orbital mechanics which also looks at the mathematics involved such as conic sections, orbital elements, types of orbits, uniform circular motion, the motions of planets and satellites, the math involved in the launch of a space vehicle, along with orbit tilt, rotation, and orientation.

Furthermore, it takes time to explore position in the elliptical orbit. orbital maneuvers, and so much more material.  Each topic has a good explanation with the math.  This might be good for a trigonometry class because many of the equations use trig.  I think I'm going to spend a couple of days just showing them these equations so they see the real life applications of tangent functions.

For classes that are not quite as high mathematically, Teach Engineering has a lovely lesson to look some of the forces and such that engineers must consider when planning to send a spacecraft to Mars.  It gives the background on what is needed, has pictures, and provides thrust for various engines.  Although it does not give as many equations, it does provide data, so students can calculate force, etc.

This would be a cool  cross curricular unit with the science department.  What do you think.

Helping Students Retain What They Learn?

One recommendation I read all over the net for students to help them retain information is for the student to read the material, close the book and ask themselves what they remember.  This is great if a student understands everything they read but in math, this particular idea does not always work.

I try to help students do this as part of my daily routine.  I always have the I can statement for the day on a board and I ask them what are we focusing on for the main lesson.  I also ask what they remember from the day or two days before.  This may be done verbally or I have them write it down as part of their warm-up.

It has been suggested students practice the skills they are taught over a longer period of time to help retention.  I sneak this into their weekly homework practice and into the daily warm-up.  I sometimes throw in percent problems, problems using fractions, or whatever topic they need the practice on. 

Sometimes, my students do not see the relation between working with simple fractions and algebraic fractions, or even trigonometric identities.  So I review the basic process of finding common denominators or dividing with fractions.

I also have several apps on my iPads for my students to use.  I love WileD Math because it has so many skills for students to practice.  I've used it in Algebra II to scaffold their rewriting standard equations to slope intercept form.  I have my low Algebra I class use it to practice solving one and two step equations, distributive property, and combining terms. 

Once students earn a certain number of points, they are allowed to spend them on any one of several games offered within the app.  I get requests from many of my students to play the game.  They really love it.

I wonder if it might improve retention if instead of starting with examples, it might prove more effective to show a problem and ask students to see if they can use what they've learned to solve this one.  After about 5 minutes, start working it out with students, using their input to solve it as a group.  Once this is done, it is time for the lecture because you've taken a step to activate prior knowledge.

I have to do this carefully with my students as they are in the habit of completely shutting down and refusing to do anything for a bit.  Its knowing your students that helps choose the best methods to use to help students retain information. 

Let me know what you think?

How Many?

How Many Pounds?

The Math of Washinton D.C. and the National Mall.

  I've been to Washington, DC a few times.  The last time, I was in the area, I visited the space and aircraft museum.  It was filled with helicopters, planes, and so many other things it would take more than one day to really explore. 

I know that Washington DC was envisioned by a Frenchman who wanted the city where everyone was equal.  The architect used the natural rises to build the important building above so they shown. 

The architect was never paid for his work and his vision did not become a reality till the early 1900's although not everything was done.  While researching the history of Washington DC, I discovered that the Mathematical Association of America produced a guide explaining the math found on the National Mall.  The National Mall is a National Park running through downtown DC.

The short 2 page field guide highlights the geometry seen along the length of the National Mall.  For instance, it mentions the geometry of water spraying out of water fountains.  It states the more spectacular patterns are due to the angle of the water coming out.  It looks at a dozen different mathematical applications found along the mall.

If you look carefully, you'll find infinity, a Mobius sculpture, a truncated trapizoid, a pyramid, a fractal and so many other applications.  Each description is short and to the point but the site also provides additional information on each highlighted feature.  For instance, if you follow the link on the geometry of water fountains, you'll find a nice discussion on the mathematics and physics of creating a spectacular display.

The explanation is from the Mathematical Tourist who used material from a paper addressing this particular math topic.  What factors control the look of a water fountain?  Its really interesting and even goes so far as to discuss which angles provide the best results. 

Just think!  If you live in the area, you could take a field trip down to the National Mall and check out everything in the folder.  If not, there is always Google Street view or Google Earth so students can check out all of the items in the Field Guide.  Let me know what you think!

Have You Used Flow Charts in Math?

The other day,  my Algebra II class started the section on solving systems of equations using elimination.  My first note read something like "Look to see if there are two of the same variables that are the same except they have opposite signs."

Rather than list the steps, I could easily create a flow chart for students to follow as they work through each process.  There would be three work flows. The first if the equations have the same variable with opposite signs.  The second if one variable could be multiplied to create the variable with the opposite sign or the third if both equations  have to be multiplied.

I created this one on a free flow chart last night so my students could use it to help determine whether they should use SSS, SAS, AAS, ASA, or HL to prove congruence.  Its not designed to help them create proofs, just something to help them decide which one to use.

I gave a couple Tuesday night at study hall and they loved it. It will help because the next activity has them determining which one they need to prove triangle congruence.  One young man thought it was cool.

Flowcharts could also be used to help solve simple one and two step equations.  I'm awondering if I could design a flow chart for trig ratio or trig identities?  Since I work with ELL students who hate to read, I think this might make it easier for them to learn the material because they don't have to do much reading.

I scribble the basic design out on scratch paper, go to the site and create it.  In addition, this is exposing them to a real word skill often used in the work place.  I first learned it when I took some computer programing classes.  Now its used in so many different situations.

I may be wrong but I see flow charts as a specific type of mind mapping which is used to organize information and material.   This is just another way to present information to students in a way they might find easier to use.

Let me know what you think.

The Math Behind QR codes

  I've noted that bar codes in the form of QR codes are invading our lives.  You see them in magazines, at the airport, on business cards, even at the airport.  I use them in my classroom when I want students to go to a specific web site so they don't waste hours trying to type in the correct URL.

Did you ever wonder how the information is stored in a QR code?  How do you read it? The Irish Times has a nice short article on bar codes and qr codes which is a great introduction to the topic.  They provide a bit of history on both topics including the original uses of both.

QR stands for quick response and can store way more data than a standard bar code.  In addition, it stores a wider variety of data than bar codes.  It states we can accept the accuracy of the reading of qr codes because when the material is coded it is coded using the Reed-Solomon error-correcting algorithm which is a sophisticated mathematical technique.


 Check out  M.E.I. who are innovators in mathematical education out of the UK have a great explanation on pg 3 of their monthly newsletter from 2012.  There is a picture with spots of color and a key to see what information is stored where.  In addition, page 4 suggests ways to use QR codes in the classroom.

 You can find additional information on structure, history, types of data encoded in QR's etc at this website.  This site does not really go into the math itself but gives a good overview with specific information on assorted facets of QR codes.

Since the Reed-Solomon error - correction algorithm is the math actually involved in creating accurate QR codes, lets look at it.  This site provides a nice introduction to the way it works but the concept is actually much more advanced than most of us teach in high school.

It turns out this particular concept is used with all sorts of storage devices such as DVD's, CD's, bar codes, etc, wireless devices such as cell phones, microwave links, Satellites, digital television and high speed modems.  In other words it has far reaching uses.

I found one activity that can be done by high school students which shows the basics of how the error correction works.  It is actually advertised as a magic trick but is still designed to teach error correction.  The seven page activity sets up code words and shows how the Reed - Solomon error correction applies to the situation.  The last incorporates math and show how the algorithm takes the information and uses it.

Any other materials I found, require more math than most of my students will ever have.  I want to teach my students about bar codes and qr codes near the end of the year when they start having summer fever.  Let me know what you think!

Math and Bar Codes

  We see bar codes all around us.  On food, qr codes, business cards, airline boarding passes, everywhere.  Have you ever wondered about how to read a bar code or the math behind them?  I hadn't until I ran across something on them and then, yes it got me to wanting to learn more about them.

To start with, how do you read the information?  What does it contain.  I found this lovely Prezi which does a lovely job of introducing UPC codes in understandable language. 


 this lovely pdf that explains how the black and white stripes convey information based on the width of the stripes.  The article starts with reading a UPC code found on items in the grocery store.  The 12 digit code contains information on the manufacturer and the inventory of the number.  You learn to read the code based on the stripes. It goes on to explain the check code in the sequence. 

The check code is used to determine if this is a valid UPC code. The total of all the numbers after being run through the algorithm must total a multiple of 10 or it is not valid.  The code reader goes through this every time a bar code is read.  The pdf includes 15 exercises so students can practice reading bar codes.

At Cut The Knot, there is a series of articles covering Bar Codes from encoding (both types), an applet which allows the student to explore reading bar codes on food, to three types of bar codes including ISBN numbers seen on books and textbooks.  This allows students to learn more about bar codes.

Finally, from Oregon State University, a work sheet on finding information from bar codes and creating bar codes.  This worksheets gives real scenarios such as the scanner goes down and the clerk has to type the code in by hand, can you find the mistake?  Or is it a valid number? 

Tomorrow I'm exploring QR codes in more detail. Although they are a type of bar code, they are enough different one cannot read them quite as easily. 

Math and Yahtzee

  I enjoy Yahtzee.  I play several games a day on my tablet.  I use a version that allows me to compete against the computer.  Sometimes I win, sometimes I lose.  I get especially frustrated when I put a 0 in Yahtzee and I roll one later in the game.  I do that because I know the odds of a Yahtzee are not very good, so I figure I'm safe.  I'm not always right.

The game was originally released in 1956 by Edwin Lowe who held the rights till 1973 when Milton Bradley bought them.  In 1983, Hasbro purchased the rights but in all those years, it has been an extremely popular game.  The game is said to be based on poker hands so there is a certain amount of probability involved with each roll.

Most of my students do not play Yahtzee.  I'm not sure most even now the game or have heard of it.  I would open with time set aside so students can play it and become more familiar with it before calculating any probabilities.  I found this nice short video on You Tube showing how certain probabilities are calculated 

The math department at Cornell University has a wonderful lesson on Yahtzee beginning by explaining the rules of the game.  It goes into the strategies for successful play before completing 8 problems asking for probabilities of different situations.  I don't know about you but I'm a bit rusty on certain types of probabilities but this lesson includes solutions and the math to get to the answer.

This lesson plan has everything needed to help students explore the probability of various combinations in Yahtzee so students learn more about calculating probabilities in general.  It is geared for 6th grade but it could easily be used above or below that grade with a bit of tweaking.

Although this is not a lesson plan the explanations for finding certain probabilities is great because it breaks it down well.  I like the way the author has explained the material.

I plan to teach this towards the end of the year and before I do anything on the actual lesson, I am going to have my students predict the odds of say a Yahtzee or three of a kind before they explore the actual probabilities.  I believe my state requires students to make predictions and carry out experimental activities.  This game makes a perfect item to create an actual experiment with predictions, finding data, etc.

Weekend Warm-up 2

Weekend Warm-up