## Friday, March 31, 2017

Most people have heard of Google Streetview and Google Maps.  Its easy to find a way to use it in Social Studies or  even English but Math can be a bit harder.

The easiest way to use Google Maps is for calculating Rate x Time = Distance because you can find the distance and the time it takes but what if you had the students look at the route using Google Streetview and asked them if the rate you calculated is reasonable?  For some of us, it wouldn't be because you'd want to stop at various stores along the way.

But what are some other ways to use these in your classroom.  If you check out Maths Maps, the author has created several activities that could easily be used in the classroom.  There are six different activities that focus on six different places and each map focuses on a set of skills.  For instance, the Madrid map works on measurement while the Adelaide map is geared for addition.

Most of these maps are created for the elementary grades but most of the things created for 5th and 6th grades could be easily used in middle school and some lower performing high school math classes.

There are quite a few ways to use these two programs in your high school classroom.

1.  Create a video of the Eiffel Tower using maps and street view to show the tower and the spot the picture is taken from.  Have students find the height of the tower and the distance to the camera.  They can use this information to calculate the hypotenuse from the photographers feet to the top of the tower.

2. Use the same information from suggestion 1 to calculate the angle of the hypotenuse line from your feet.  With that information, they can calculate the other angle from the top. If you want, they can calculate the trig ratios.

3.  Take a picture of the Roman Coliseum.  Draw the length and width on it so students can calculate the basic equation for its ellipse shape.

4. Find the huge wheel at Canary Wharf in England.  Research to find the radius of the wheel and calculate the area, or equation of the wheel which is a circle.

5. Find the slope between two places in Switzerland on the Matterhorn using the information from google maps and the internet.

These are some beginning suggestions.  I am going to present on this topic at the Kamehameha Schools Educational Technology Conference the beginning of June.

Have a good day everyone.

## Thursday, March 30, 2017

### Musical Scales and Math.

I've been conducting testing this week and got just a bit behind.  At night, I've been watching Numb3rs and the episode I saw last night mentioned musical scales and math.

The comment suggested the pentatonic scale is based on fractions. When I heard that, my mind went wow! and cool! at the same time.

I have students who love music and this might be a way for them to become more interested in math.

I mentioned this to a friend who has a good background in musical theory and he replied oh yeah to find the next note in the modern scale you multiply the frequency of the note by the 12th root of 2.  That totally blew my mind because I didn't realize it was that mathematical. They've established the note of A is 440 Hz.

It turns out Pythagoras, the man who formulated the Pythagorean Theorem, stated that pleasing sounds come from frequencies with simple ratios.  If you play music you've heard of octaves  with a ratio of 2 to 1, perfect fifths with a ratio of 3 to 2, a perfect fourth is 4 to 3, and major thirds with a ratio of 5 to 4.

This means that if an A is 440 Hz, a perfect fifth would be 660 Hz because 660 to 440 is a 3 to 2 ratio. This is a cool idea that certain types of notes are based on ratios. A perfect third would be  550 to 440 Hz while an octave is 880 to 440.

Another way to look at the ratios is based on the waves.  The ratio for E to C is about 5 to 4 or every 5th wave of E matches up with every 4th wave of C.  The actual ratios are approximate in reality and are as follows.
1. Middle C which is considered as a whole.
2. D has a ratio of 9 to 8 to middle C
3. E has a ratio of 5 to 4 to middle C
4. F has a ratio of 4 to 3 to middle C
5. G has a ratio of 3 to 2 to middle C
6. A has a ratio of 5 to 3 to middle C
7. B has a ratio of 17 to 9 to middle C

This is a new use of ratios and fractions I have not known about before.  Next time I teach ratios, I'm going to include this material.  Let me know what you think.

Now it is not always this simple but its a

## Wednesday, March 29, 2017

### Height and Steps

The other night I watched an episode of the Librarians.  The girl with something in her brain who could see all her senses input made a comment about telling the size of a set based on the number of steps the actor takes to cross a set if you know the actor's height.

While researching this comment, I came across a really interesting article in Scientific American on estimating a person's height from the length of their walk.

The article began with discussing some interesting ratios I hadn't realized.  For instance, if you have your arms completely outstretched, the distance between the tips of your hands is about equal to your height. The length of a person's legs is related to their height as a ratio.

One of the activities you can do to find it is to measure out 20 feet in a flat area like a hallway or cement walkway.  Mark the beginning and end.  Measure the height of each walker.  Have them walk the 20 foot length. Count the number of steps it takes them to reach the end.  Determine the length of their stride by dividing the 20 foot length by the number of steps it took them.  Then divide the stride length by their height, both numbers should be in feet.  The answer for all walkers should be about .40.

So if you have the length of a person's stride, you can divide it by about .43 to get their approximate height. The answer won't be exact but it will be close.  .43 is considered to be an average.

Other interesting ratios include:
1.  You are about 1 cm taller in the morning after sleeping all night.  You shrink a bit during the day.  Imagine being able to have students measure themselves first thing in the morning and last thing at night.  Students could calculate the 1 cm as a percent of their total body height so they'd know their "shrinkage factor."

2. The ratio of the femur bone to the height is interesting.  It is about 1/4th your height.  You could use this to discuss how forensic scientists determine the height of the person whose bones were found.  A practical use.

3. Another interesting ratio is the head to the body which turns out to be change depending on the age of the person.  A small child has a 1 to 4 ratio while an adult usually has a ratio of 1 to 8.

Finding these ratios could be easily done in the classroom by having students carry out an experiment of two.  Scientific American has two different pages with everything you need to have students learn about these ratios themselves.  One is the Human Body Ratios page while the other is Stepping Science finding the height of a person based on their stride.

These exercises would be great when teaching about ratios because these ratios are ones they can easily relate to.  Let me know what you think.  I'd love to hear from you on this.  Thanks for reading.

## Tuesday, March 28, 2017

### Weather Math

Most of us check the weather report most days.  We want to know if its going to be rainy, snowy, windy, or sunny.  I tend to check the weather report when I'm due to travel because if its bad, I won't be able to get out of the village.

Yes, it has happened when a storm blew in faster than expected and visibility dropped to nothing.

The type of math used in predicting weather is called numerical weather prediction.  This is actually a branch of atmospheric sciences and was pioneered since World War II.

This type of math really took off in the 1980's when computing power reached a certain level.  In addition, accuracy has improved with the better computing abilities.

Numerical Weather Predictions is composed of equations, numerical approximations, boundaries, domains and a couple of other things.  What is most interesting are the equations they use in weather predicting.

1. Conservation of Momentum - 3 equations
2. Conservation of Mass for both water and air.
3. Conservation of Energy using the first law of thermodynamics
4. The relationship among density, pressure, and temperature.

The form of the equations vary slightly due to where in the world weather is being predicted.  Wind patterns are different, humidity changes, pressure changes slightly due to elevation,  and other factors.  In addition all equations have to be converted to algebraic equivalents because computers can only do arithmetic, not calculus.

In addition, Reynolds Averaging is used to separate out the resolvable and unresolvable scales of motions in the equations themselves.  This is accomplished by splitting the dependent variables into resolvable (mean) or unresolvable (turbulent) components.

If you noticed both physics and numerical calculations are heavily involved in predicting the weather.  There are more factors involved in this process than I mentioned but if you check out
this presentation, it gives a good explanation of Numerical Weather Predictions and provides some excellent detail.  It shows the actual math and provides detailed examples of all facets used in the process of predicting weather.

## Monday, March 27, 2017

### More on Transferring.

It is well known children struggle to transfer what they've learned to other situations.  If you read Friday's entry, you know one thought is we teach clean neat math in class but real applications are a lot messier.

Students often have difficulty figuring out how their prior knowledge applies to the new situation.  They have trouble recognizing cues such as if we looked at a pay as you go plan with a specific price for a certain amount of data.  The equation might be \$35 for the base + \$5 per half gig of data added or 35 + 5x = the amount.

This is a simple linear equation but my students won't realize that because all the problems they've seen are only symbols rather than seeing a context.  They don't know how to relate the general equation to the specifics.  I will be the first to admit, I do not teach it the way I should.  I've only recently started looking at specifics so I can change my teaching to help students learn cues.

According to something I read, it is important to use the "I do, we do, you do" type of teaching, also known as model, guided practice, independent practice.  I know I should be doing this more but it always seems like I have to leave the classroom to help test students during one of those mandated tests we have to administer.  You have to practice anything to master it.

It is also suggested, the instructor assess how students are doing on learning to transfer knowledge but do it without grading.  This allows the teacher to determine what needs to be done next.  One way to do this is to give them questions in an unfamiliar format with no cues on what it relates to.  Review how they attempted the problem and analyze that to see what the next step is.  Were they able to figure out the type of task?  Did they choose the correct tool?

Finally, change the set-up so they see that prior learning appears in many different forms.  Research indicates they need to see the change in setting, format, context, and language so students are more flexible in their thinking and accessing prior knowledge.

Tomorrow I'll finish off this topic.  I'm spending three days on it because most curriculums do not spend enough time on building transfer so as a teacher, I think its important to learn what we can do to improve student transference.

Let me know what you think.

## Friday, March 24, 2017

### Transfering Knowledge From Math to Other Subjects.

Math is one of those subjects which provide a foundation for science, engineering, statistics, and other professions.  It has been noticed that students often are able to perform the mathematics in their math class but when required to perform the math in another subject, they cannot do it.

I've seen it myself.  When students are required to perform conversions such as inches to feet or centimeters to meters in science they struggle but they can do it in math.  They even struggle when trying to solve a linear equation in science they easily solved in math.

There have been several studies on why students have difficulty transferring the knowledge between the math classroom and other subjects.  The results are quite interesting, especially as factors can start as early as pre-school.

We all know that mathematics has its own language which can be quite specialized or at least have meanings different from conversational use.  One researcher discovered a better indicator of success is the amount of math vocabulary a child has.  If a child does not understand basic words such as "plus" or "times" they have difficulty learning math.  In addition, with out the basic vocabulary, they have difficulty expressing the answers.

Its also been discovered the way a concept is presented can improve a students ability to know when and how to apply it to a situation.  The type of practice does make a difference. Rather than focusing on the symbolic practice, expand it to problems which help students see the underlying relationships among the numbers.  In other words, if you are making paint and you change the ratios of the basic colors, you change the final color.  You would have students use a simulation to practice doing this so they see how the changes in the ratios, change the color.

Another observation boils down to the way students practice the material in "clean" situations while most of the time, the applications are actually "messier" and they don't know how to move from the clean to messy situation.  The clean situation is really just drilling while the messier is application.

It is recommended that teachers teach and test understanding when applied to various situation to help transfer their knowledge. In addition, it is suggested the teacher establish and highlight the goal of transference to the students.  It is important to visit and revisit the goal.

Furthermore, students need to judge what skills should be used under which situation so they can transfer knowledge. One way is to model think alouds so students observe the process in action.  Then have them practice this skill with immediate feedback so they learn how to apply their knowledge to a situation.  This helps develop their transfer knowledge skill.

This is focusing on one specific technique to help develop transference of knowledge.  I'll touch on more later on next week.  I hope you all have a good day.  Let me know what you think about this topic.

## Thursday, March 23, 2017

### Math Relay Games

Do you remember when you were little and in school, you'd participate in those math relay games?

I remember one where the class sat in rows.  Each person in the row had an equation.  The teacher called out a number for the first person in each row to use.

As they finished the problem, they'd pass the answer off to the second person who used the answer in their problem. This process would repeat until the last person finished and raised their hand to have the teacher check it.  The first row with the correct answer at the end, won.

The other night I thought I might want to use it in class with variations on it but I only know the old fashioned way with 3 x 5 cards.  These are fairly easy to prepare but I wondered if I could have students use something like snap chat to create relay games.

I don't know much about snap chat but I do know my students love it. What if I assigned a problem to each small group of students?  The groups of students worked on the problem either on a whiteboard or paper, and when done, snap a picture of one member with the answer, they could shoot it to me via snap chat or other program.

Even better, they could work together on a collaboration program or with one of the google suite so they work together on the program and they can post the solution showing the work.  Everyone I speak to, says its important for students to learn to work together or collaborate.

The nice thing about using a digital program or device is there is always a time stamp to prove who was first.  Before, when relying on the naked eye, it was difficult to tell who finished first if two or more hands shot up first.  In addition, students would sometimes raise a hand, even when not done so they were not left out.

It would be possible to run the old fashioned relay races by having students text their answer to the next person to use in their problem.  The last person text's the answer to the teacher.  This again provides a time stamp for who got the answer in first.  The teacher does not have to announce the winner right away but could text a reply stating the answer was received.

I would love to hear from you readers about your thoughts on this idea for a relay race or for working together in groups. These are thoughts at this moment and the feedback will help me fine tune the idea.  Thank you ahead of time.

## Wednesday, March 22, 2017

### Time Zones

I just spent about 8.5 hours flying from Philadelphia Pennsylvania to Anchorage, Alaska a span of four times zones.  It is exhausting because of having left so early in the morning and arriving at my destination mid afternoon.

Its interesting that the world is divided into a minimum of 24 time zones based on the idea that each time zone is 15 degrees from the next time zone or about an hour apart but in reality it does not quite work that way. There is the GMT line or Greenwich Mean Time, the International Date Line which have added a couple of extra time zones to everything.  Then there are a some places such Singapore or  North Korea as which only have 30 or 45 minutes in the time zone.

One large country, China, only has one time zone. China has operated on Beijing Time or Chinese Standard Time since 1949 when the communists took over.  A question asking about travel time in China would require no additional time zone calculations but if you flew to Singapore, you'd have to keep in mind the 30 minute time zone.

There are several sites which provide some very good time change problems complete with explanations and very real problems.  For instance, Space Math by NASA takes time to explain the time zones in the United States but its problems deal with what time astronomers need to be ready to watch a solar flare.  I've actually done some calculations like that to determine if I could watch a solar event.

Berkley has a nice set of problems which take this a step further by involving more countries after having students practice finding times when going from one time zone to another.  The questions require students to calculate differences between Central Australia and Alaska or Universal Time and California.

I was unable to find problems in which the traveler began in Germany and ended in one of the countries with a 30 or 45 minute time zone.  I think it would be cool to have students create a a trip through certain countries with information on time zones, take off and landing times for a realistic activity.

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

## Tuesday, March 21, 2017

### 9 Common misconceptions.

While researching yesterday's topic, I stumbled across a list of mathematical misconceptions some of which I've had students happily share.

I'm sure you'll recognize some or all of the misconceptions listed below.  I'm also sure some will make you smile at the memory of a teacher telling you that exact thing in elementary school.

I know, I heard them myself.  So here is the list.

1. Three digit numbers are always bigger than two digit numbers.  This rule comes about because when they first learn numbers, they are only exposed to whole numbers.  In that case, this rule is correct but once decimals are thrown into the learning, it no longer applies.  3.24 is not bigger than 6.2.

2. When you multiply two numbers together, the result is always larger than either of the original but that is only true with whole numbers.  Once students begin using fractions or decimals, this may not be true.  one example is 1/2 times 1/6.  The result, 1/12, is smaller than either one.

3. Often students think the fraction with the larger number in the denominator means its larger such as in 1/4 and 1/8.  They sometimes think 1/8 is larger than 1/4 because 8 is larger than 4.  I think this has to do with 8 is larger than 4 normally with what they've been taught so when the context changes their understanding does not.

4. Most students see two dimensional shapes in only one orientation such as a triangle with the base always at the bottom part of the shape rather than placing it at the top with the vertex pointing downward or off to the side.  Teachers need to change the orientation so students do not get in the habit of seeing it one way.

5. In squares the diagonal appears to be almost the same length as the sides and students may assume they are the same.

6. When multiplying by 10, simply add a zero.  This works for a whole number but not for a decimal number.  You could add a zero but it does not help you to remember to change the position of the decimal.

7. Ratios where students get used to comparing one object to another such as two carrots to three peppers rather than looking at two carrots to five vegetables.  When the situation comes up where they need to set up a part to a whole, they often have trouble.

8. Students often confuse perimeter to area because they count squares for both of them without understanding the whole square inside the shape is counted for area while they are only counting one side of the square for the perimeter.

9. Students often have difficulty determining the scale used by the measuring item. Not all scares are divided into 10's. Many students do not count the markings to figure that out, they assume its always going to be 10.

I understand why students are taught many of these rules when they are in elementary school but it does a disservice teaching these are "rules".  Students need to to quit learning "rules" which only apply to a narrow population of numbers.  Hopefully, teachers will quit doing these so students are more open to learning new situations.

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

## Monday, March 20, 2017

### HIgher Education.

I had to go to a family gathering over the weekend.  I spoke with one who is currently working on his PhD in Chemical Engineering.  He is working with polymers in an interesting way.

He and I discussed the skills he needs in his line of work.  It came out he doesn't bother keeping track of certain chemical interactions because he looks it up anytime.

We also discussed when he needs to do any type of data analysis, he has programs to complete the analysis.  He does not worry about remembering various formulas.

He stated, it is more important for him to know how to use these programs and interpret the results than it is to remember how to do it by hand. I found that interesting because the school system is still way behind this belief.

It does emphasis the idea that math provides answers and its important for students to interpret the results they get from their calculations.  I don't do this enough.  I teach students how to solve equations but I do not take the extra time to ask them to interpret their answers in terms of the problem.

I have them solve one and two step equations but I do not take the time to discuss what the answer might represent. When I studied math in high school, it was only important to solve equations, not understand anything about the meaning of the results.  That was not important.

According to current thinking, it is important for students to be able to create mathematical observations about their solutions.  They need to connect the mathematics with the situation.  One facet of a mathematically proficient student is their ability to interpret results in context of the situation and reflecting if their solution makes sense.

They are able to take this reflection and make changes to create a model which is closer to what it should be.  This is a real life process.  The young man, talked about using the results of the data he's collected to determine what the next step should be.  He adjusts factors, tests, and recalculates.

Having students work on performance tasks which require them to examine their works to find tune their assumptions is important and used more in life than having a problem done once and accepting the answer is done so you have nothing more left.

Yes, it sounds like a science experiment but if you are creating a mathematical model of a situation, it often takes several tries to get the right equation.  It seldom happens immediately with the first try.  Its important to create a situation to help students  create models which take several tries to get right.

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

## Friday, March 17, 2017

### Formulas

I assume most of you have to teach students to rewrite literal formulas in isolation because that is one of the standards we are required to teach.

What I don't understand is why that is necessary.  It makes more sense for most students to connect the literal formulas to real life use.

Lets face it, r*t=d and I = V/R are just a collection of letters to most people until its put in context.  We use literal equations all the time but not without values.

Most of us select the literal formula for the appropriate situation, substitute values to find the answer for the missing value.  I don't know of anyone who rewrites literal equations just for the fun of it.  Is it really important to rewrite I = V/R to I*R = V?  Isn't it more important to have students substitute values before solving?

I don't think of rewriting the equation, I think of solving the equation with variables.  There are now calculators out there where you type in the values and the answer pops out without doing the calculations.

Why is this considered an important skill?  Why do we make students rewrite the literal equation in all its ways rather than focusing on showing you are solving a one step equation.  If we expect students to be good in mathematics, we need to provide more connections and more real life applications of what we are teaching.

I'm not even sure why this particular skill is still in the standards.  I wonder if it is there due to people who have a fond memory of doing this in school.  I thought it was a waste when I took math in high school and we are still making students learn this even though they can just find the missing value.

Is this necessary?  I don't think so.  I think its time to get rid of this particular standard and focus on more important things.

Let me know what you think.  I'm in transit till Tuesday.

## Thursday, March 16, 2017

### Animation.

Sorry, about yesterday but my plane was 2.5 hours early and I had to rush out before finishing it.  So you get it today.

One of the math classes I'm teaching this semester is an animation class based on Khan Academy's Pixar in a Box.

I say based because I'm using the online material provided with the videos and the practice activities but I added in a more in depth math component.

As we work through the package, I include instruction on the actual math associated with the lesson.

When they did the section on animation, I integrated more instruction on linear interpolation.  For the character modeling, students learned about all the different ways you can find weighted averages in real life.

Right now, the students started simulation of the hair. Specifically, the hair of the lead character in Frozen.  I've never seen the movie but my students have.  They've learned her hair was simulated by looking at the workings of springs because springs bounce the same way as naturally curly hair works.

So as part of the lesson, they are learning to use Hooke's law.  Someone one, I know, asked why I was teaching science in my math class. I pointed out Hooke's law is a mathematical equation which students can learn to solve for force, the constant, or distance.

Its interesting the separation of subjects is found even among teachers.  You can't solve a formula or equation without doing math.  Math and science go hand in hand.  I just introduced rewriting the formula to find distance or the constant but its a real life application of math.

I really like exploring the math in more depth so students see exactly what the math is that is lightly touched on.  This prepares students for completing the second part of the topic which focuses more on the math but doesn't always teach the details.

Many of the students in my animation class struggle with math or are under motivated.  This class gives them a reason, a real reason, to learn mathematics.  They love playing with the animation activities but accept they have to learn the math.

I'll keep you posted.  I am happy to use Pixar in a Box because its all set and ready to go.  The only issue I have is that some days, the internet slows to a crawl.  A total crawl and only half the kids can be on at any one time.

Have a good day and let me know what you think.

## Tuesday, March 14, 2017

### Look it up!

I've been reading quite a few books in which the author states that it is fine for students to look up the information they need to answer what ever question you pose.

I've recently tried it in my classroom.  For instance, I asked for the volume of a rectangular prism in Geometry.  When students asked me about the formula, I told them to look it up.

It is in their notes but they are more likely to look it up than refer to their notes.  In addition, I can ask a question such as how are inverse functions used in real life.  They can find the answer.

If we look at the world out there, it is no longer factory based.  It has become more digital and we use the internet to find things.  I use it to find videos to use in class, worksheets and problems which are in digital form which can be posted.

So is this a skill we should be cultivating in the Math classroom?  Will it help students develop number sense, especially if they rely on an assortment of on-line calculators to do the work?

I fight with letting them use a calculator to find the answer vs one which shows the steps of how the answer came about.  My personal belief is if they see how it is done, they might develop enough sense to know their answer is close to right or not.  Most of my students feel that if they put numbers in, the answer is automatically correct even when they put in a wrong number. They are often startled when I look at an answer and tell them, they made a mistake.

They want to take every answer and have it in either whole or decimal form.  I am not sure how to impress them with the idea of needing an exact answer vs an approximation.  In real life, we do not often say a circle has a circumference of 3pi.  We usually say its about 3.42 inches, feet, or which ever unit it is.

Is it important to be that precise for most people in today's society?  I know in some fields, yes but for most people?  I struggle with it because most of my students will not ever need to be as precise.  I think it is important to teach students to be more independent learners who are capable of finding and learning information off the internet, even in math.

Most of us, unless we use the math ever day, have to look up a formula or how to apply a formula to a situation. I think this is the skill we have to teach even if we teach traditional math.

I'd love to hear from you with your thoughts on this topic.  Thank you for reading.

## Monday, March 13, 2017

I'm recently implemented google classroom in all my classes for several reasons including the fact I have a classroom load of iPads, I'm not using enough.

My first step has been to list all warm-ups, assignments, due dates, and announcements in each period.  I discovered I can share Khan Academy videos with students via a Google classroom share button.

This week, I've taken things a step farther because we have just about run out of paper and we cannot make as many copies as we have in the past.  They've asked us to cut down as much as possible.  So this week, I'm creating assignments with problems in google classroom so students do not have to keep track of as much paper.

I have not set up google docs, slides, sheets, etc yet because I will not get them on my iPads till next fall according to the tech department.  So I am either writing the problems into the assignment or attaching screen shots of the problems I want done.

I don't know how its going to go but we'll find out Monday when my students log in.  I hope it cuts down on the issues I've had with the athletes who travel.  Most do not get their work done, loose it, or any one of a number of other reasons.  They can log in via the internet while they are gone and can keep up.

I hope over the next 9 weeks of school to place more and more on line so I have fewer papers to grade and make them more independent.  I put up links to pages they can explore with videos or with written examples.

One thing I do to help them learn to take notes, is to play a video, stop it so they know what types of things make good notes for later on.  I'm hoping next year, I can just post a video for them to take their own notes.  Hopefully by next year, they will be much more independent in their learning.

I will continue to post updates.  I should also tell you, I anxiously await the publication of a google classroom book which is focused on Math.  I can hardly wait till April.  One of the authors is Alice Keeler.  She is the one who inspired me to take my first step into using Google Classroom and I thank her.

I'd love to hear from others about using google classroom in math and how's it going.  Thanks for reading.

## Friday, March 10, 2017

### Composite Functions in Real Life

I am currently teaching composition of functions in my Algebra II class.  My students are still learning it and have not gotten around to the question of when it is used in real life.

The thing is, I've never been told how to use it in real life.  I know I use it to determine if two functions are inverses of each other but I have no idea otherwise.

I read of a beautiful example showing a f(g(x)) that cannot be done in the reverse.  You are bottling soda in bottles.  You have one function which fills the bottle while the other function puts the cap on.

So where in real life do we see composite functions used?  Quite a few places it turns out.

1.  Predator - Prey situations where there are outside influences such as a virus or another predator.

2. The population living near the coast effects the number of whales which effects the amount of plankton available.

3.  The salary based on a commission for anything sold over a minimum amount.

4. Calculating the cost of life insurance based on age.

5. Buying something, paying tax, and paying a delivery fee is another example of composite functions.

6. A circle whose radius increases over time.

7.  A vehicle uses so much gas per hour and there is a cost for the gas.  So the two functions combined are the composite functions.

8.  The duration of a cruise  as a function of the the speed of the river.

So basically anything that is based on something else tends to be a composition of functions and is all around us. So many possible examples.

I love finding out we do use composite functions everywhere in life.  Let me know what you think.  Have a good day.

## Thursday, March 9, 2017

### Gear Ratios

The other day when I wrote about ratios in game design, Adam Liss reminded me of gear ratios.  Thank you for this entry.

To begin with gears can transfer motion, slow things down or speed them up, create changes in torque, and control  motion.

Gears and gear ratio play a huge part in most bicycles with more than one speed.  I have one with like 15 gears but I seldom use more than 3 or  4 depending on the hills I have to go up.

The first thing to consider is the gear ratio which is the ratio of teeth on each gear.  Which means if you have a gear with 8 teeth and another with 40 teeth, the gear ratio is 40/8 or 5/1.  This tells us the smaller gear will rotate 5 times to one time for larger 40 tooth gear.  It also means the smaller one will rotate 5 times faster than the other.  This type of set up is designed to increase torque.

On a bicycle, its easier to see gears because of its design.  On a bike, the gear ratio is also referred to as the velocity ratio.  Usually the pedals are attached to the largest gear while the back week has several gears of differing sizes attached to it.  When you pedal, it moves the wheel which moves the gears and the chain.  The more teeth the back gear has, the easier it is to pedal up hill.

A racing bike is normally has 52 teeth on the front with 13 teeth on the back so its a 4 to 1 ratio while a regular bike might have the best gear ratio of 44 teeth to 16 teeth or a 2.75 ratio.  It all depends on the type of bike and the type of gears installed.  Many bicyclists will customize their gears based upon their needs.  I've got a hybrid which will go off road or on road if needed but I don't know the gear ratio only because I never bothered asking for that information.  I gave the bike person my request and she put it together.

If you look at your car manual you might find something about the pinion gear found on the motor of an electric car, and the spur gear found on the drive axle of a car.  So if your pinion/spur gears is 18:90, although they usually use the spur/pinion gear ratio of 90:18 or 5 to 1.  This is similar to the example. There are also transmission ratios, final drive ratios and other ratios associated with cars.

If you do a bit of looking, you find out that first has a ratio of about 3.166 to 1, second is about 1.882 to 1 while third is about 1.296 to 1 and fourth is around 0.972 to 1.  It would be easy to have students calculate the number of teeth by calculating the actual ratio with whole numbers.  This would be doing a reverse calculation.

To figure out the actual ratio, it is recommended you take apart the car enough to find the actual gears, otherwise car manuals tend to only list the reduced ratios.

Thank you to Adam again for this idea.  Have a great day everyone.

## Wednesday, March 8, 2017

### Partner Tests.

Over the past year, I've been moving away from the old testing methods to one that has students working together on a test.  You might wonder why?  I'm finding it allows students to learn more of the material before we move on.  It also fosters collaboration, and communication.

It is well known that many students know the material but when they have to take a "Test" alone, they panic and often forget much of the material.  The grade does not always represent their knowledge.

If students are allowed to work together on a test, it cuts down on their anxiety.  In addition, a well written test can foster collaboration and communication.

Furthermore, students who work together on a test often implement peer tutoring to help each other because one explains to the other how to do something or why their way is wrong.

I have used two different types of partner tests.  The first is where the tests have different problems but they share the same answer.  This does not work as well when you do factoring or other where there really is only one answer.  It works for solving one and multi-step equations, finding slope, etc.

On this type of test, I tell them if they get the same answer, chances are they are correct, otherwise it is a good idea to check their work because one or both might be wrong.

The other type of test is where I allow both students to work together.  I try to pair students who will both work on the problem even if one is better at solving it.  I do not pair students where I know one will only copy the work.

When I use the second type of test, I expect both people to show their work on their own paper but I never give partial credit on this one.  The answer is right or wrong since two people are working together.

I noticed in my class with lower performing students, many of them got the hang of linear interpolation when before they struggled with it.  I allowed them to use their notes to help them figure out the next step.  The serious ones utilized the opportunity to help them.

I realize testing is a form of assessment but I don't see why we can't use it as a way to promote learning.  I hate regular tests because I'm the one who freezes, even when I know the material.

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

## Tuesday, March 7, 2017

### Game Design and Ratios.

I downloaded a game which is designed to help students learn to work with ratios but I'm not sure its a good one.  The premise is the student is given a ratio such as three greens to one red and they have to connect them together with a finger drawn line.

I played it but it seemed just like any other fancy game. I'm not sure how it teaches ratios to students.  But ratios are often used in game design itself.

One type of ratio is a fixed ratio which is where the game designer sets up a ratio after so many repeated actions.  An example might be an extra charge after collecting 20 jewels.  This happens all the time and doesn't change, thus its fixed.

This ratio leads to a period where not much happens until the player decides to go after the goal.  Then the player will move as fast as possible to achieve the goal and getting the reward.

The other ratio is a variable ratio, where the number of action changes so it requires a different number every time to achieve a goal.  For instance, the number of enemy fighters shot down to get a new fighter of your own, changes each time.  The variable ratio encourages more regular activities without the pause associated with the fixed ratio.

Not all rewards are on a fixed or variable ratio.  The designer might choose to use a fixed interval where the player receives a reward.  It might be one reward for every hour played.  It stays the same.  Where as the designer might choose a variable interval where the time played varies before earning the reward.

In a sense the intervals are also a ratio.  Instead of 1 new protection for every 15 wizards dispatched, it might be 1 new protection every 20 minutes.  Its a ratio using time rather than objects.

In addition, the golden ratio plays an important part in game design.  The golden ratio (1:1.61) is used to create proper looking surroundings for the game itself.  So by applying this rule when creating the background, trees, etc, you get a more realistic look because this ratio is found throughout nature and we are used to seeing things that way.

The golden ratio is often used in by dividing the scene into thirds to get a rough idea of where to put things.  In some games, the first third line represents the horizon while the second third is the architecture line.  Other games use the first third line as a line while the second third is the eye height line for the characters.  The vertical lines express where the enemy stands and where the hero or your character stands.

Again ratios.  I love that ratios are found in game design because there are so many resources out there to teach students game design.  What do you think.

## Monday, March 6, 2017

### Draw 3 D Junior

The other day, I stumbled across a very nice app for students aged 6 to 12 called Draw 3D Junior: Learn Geometry & Create 3D Models.

This app does cost \$2.99 but it is different from many other geometry apps in that it teaches students to draw 2 dimensional shapes in a world where you can move the shape around and examine it in 3 dimensions.

The  app offers three options beginning with the learn section, create, and browse.  The first option, learn, takes the student through 30 different polygons from triangles and squares to 3 dimensional shapes such as cylinders and spheres.

The app is set up so the student works through the shapes slowly.

The app leads the student through creating the shapes step by step by causing the dots to shine.  This way it is easy to follow the apps lead.  At the end, the student has created the shape.

The shape moves around as each segment is put in so the student see all sides.  When its done, it moves around or you can move it around to examine all 3 dimensions.

In addition, once the polygon is finished, the app shows the name of the polygon, the number of faces, vertices, and edges.

According to the app, the tetrahedron has four triangular faces, six straight edges and four vertices.

The create allows the student to make their own design.  I made a house but it took a bit of maneuvering to get it right in all three dimensions. I had to move the perspective around a few times to make sure segments were the correct length.

The browse feature shows you other shapes which have been made.

This app could be used in high school for students who are quite visual and need to see the shape from various directions.  It would be appropriate in an English Language Learner classroom, or with students who need extra scaffolding.  I actually enjoyed playing with it.

## Friday, March 3, 2017

### Airplanes and Math

When ever I get ready to travel, I call the local agent to find out if they know when the plane is coming.  This leads to a yes its due at ________ or call back in 30 minutes when we hope to have an update.

If you get the green light, they will ask you how much do you think your luggage weights.  No they don't weigh it because you go out to the airport when your plane is coming in for a landing.  As soon as it comes to a stop, you haul your stuff out and wait till your name is called.

The other thing they always ask is "How much do you weigh fully clothed."  In summer you add about 5 lbs but in winter you throw in 15 lbs to cover your winter gear. This is important because a planes performance is determined by its weight.

There is a whole set of calculations for figuring out how much baggage the plane can carry.  All airplanes have a maximum weight they cannot exceed or they cannot take off safely.  The total weight includes the weight of the plane, fuel, passengers, baggage, etc.  They weigh the checked baggage but not the carry on pieces.  My local one weighs everything including all carryons.

Since most regular airlines do not require people to weigh themselves, they use a constant weight for men or woman and another set of constants for clothing in summer or winter.  These are used to calculate the weight of the pilots, stewards, and passengers.  There is also all the luggage associated with travelers. This is why sometimes you arrive on one flight while your luggage arrives on another.

When they calculate the weight of the fuel, the number depends on the type of fuel used.  A certain conversion factor is used.

It is quite easy to find all the necessary numbers for any airplane on the internet so students could create a spread sheet or do a project where they are responsible for determining  the maximum amount of luggage which can be loaded. Everything to do this project is found on the internet.

Let me know what you think.

## Thursday, March 2, 2017

### More or Less

I remember growing up and being assigned the requisite 20 to 30 problems per night. The problems were never the ones with an answer in the back so you couldn't check your work.

I know the thought behind this is simply, if a student has trouble the more practice they get, the better they'll get.  Unfortunately, this does not work very well.

If the student has no idea how to do the problem, they will probably not get the hang, no matter how many times they do the problem.  If they can't check the answer, they have no idea if their final product is correct.  So if they learn it wrong, they will continue doing it wrong.

In the past, I have left the teacher's edition open on a table so students could check their answers.  In the upper level classes, students would often ask me questions if they did not do it correctly.  The younger ones, often copy the answer without showing the work but I found some of the calculators out there online will list steps so students can see the process.

I like something said at the recent technology conference.  It was suggested you assign way fewer problems but have students include a written explanation of what they did rather than just solve it.  The explanation provides a form of assessment.  It allows the student to share their understanding of the process.

I know, some people believe its enough for students to show they can solve a problem but I don't think so.  I think its important a student is able to explain why they are doing a certain step.  Can  student explain why they want to isolate a variable.  Why is it important?

Does a student understand when they must follow the reverse of the order of operations and when its not as important?  Can they explain when you must do that and when you don't have to?  Can they explain why they chose to solve a certain problem using proportions?  Does it matter if you divide before you square something?

So many times math teachers expect students to do certain steps because it was the way they learned it.  I read a book which made it clear when you are solving a problem such as 5(x + 2) = 20, you can divide both sides by 5 first rather than distributing it first.  I'd never though about it because I was taught to distribute first before trying to solve.

I am starting to offer students the opportunity to answer fewer questions as long as they explain what they did and why.  So far, no takers but I'm sure as time passes, I'll get them moving that way. I'd like to hear what everybody's thoughts on this.

## Wednesday, March 1, 2017

### Factoring Trinomials.

I have been trying to find a way to clarify factoring trinomials with a leading coefficient of other than one.  When I first learned it, you had factor the leading coefficient and the constant, figure out which set of factors you needed before finally finding the answer.  It took quite a bit of time and frustration.

Several years later, I learned to use the diamond with the product at the top and sum at the bottom to find the factors.  It worked well but my high school students often could not relate the numbers in the diamond when rewriting the equation into four terms.  The other day, I saw a video which used the diamond method but with a slight difference.

When he wrote the product, he included the x^2 and when writing factors, he wrote it with the x instead of just the coefficients.

This was so much clearer because anyone could see the relationships and its much easier to see where the factoring terms go.  I showed a write-up with a few more steps of what the person showed.  I had the students take notes from the video with added steps and comments.

Today when we worked the first problem, together.  I insisted they look at their notes while we worked the problem.  I am taking steps to make them more independent learners and this includes learning to look and follow their notes.

One of my students said this way was better because it reminded him to include all the variables.  I think I'll be teaching it this way.  Let me know if you have a good way to teach this topic.