Google Classroom.

  Last week, I attended a technology conference where I spent a whole day on google classroom.  I've been thinking about what the presenter said and it made sense.

I have students who are gone traveling for sports, 2 to 3 days a week.  They often miss a lot of material when they are out of the classroom.  I decided that google classroom is a good tool so I can list all the warm-ups, assignments, up coming tests, etc and they can check it out while they are gone.

I've already discovered I can create a quick question for students to answer.  When they turn in their answer, I can reply and grade in a short time. I am able to list assignments, due dates, start discussions, etc with no problem.  If students get stuck while traveling, they can post questions to me via private conversation.

Our school is set up as a google education school which is one reason I chose it.  I've already been posting links for videos so if a student gets stuck, they can watch the video as many times as needed to help finish work.   I have assigned a project to my geometry class with all the information and due date so students can check and recheck it.

I still give out actual papers with work on it but as soon as I have them adjusted to using google classroom, I'll start telling them to answer 5 problems but they have to show all their work and explain what they did and why they did it to show me their understanding.

My first period is doing well adjusting to it.  They come in, sign in, and start on their warm-ups.  In fact, several students made the suggestion I label all questions as a warm-up to  make it easier for them to find and complete.  I love the suggestions.

My next step is to have the google apps put on my iPads to open up more opportunities for student work.  Eventually, I would like to have it so students use little if any paper, otherwise they loose everything including their calendars, notebooks, and homework.  I want to make life a bit easier.

So far I am enjoying this tool.  It does not take me long to grade warm-ups, set up work for the next day, or make announcements.  Yes. I am moving into the 21st century.

Math and Pilots

  As most of you know, I teach in the Bush of Alaska.  That means, out where I am, you cannot drive on a road anywhere.  The only way to get to and from most other places is via airplane.  Small 6 to 9 seats per plane which haul freight, mail, and people.

Is amazing how much math a pilot actually uses even with the technology we have today.  On the way back to the village, the person in front of me showed this huge book full of charts, graphs, and information on how long a landing strip is needed based on speed and temperatures.  Head winds can cause a plane to use more fuel because it is is being slowed down or a tail wind can make the trip quicker using less fuel.

Most pilots use headings which are in degrees with a speed or a use of vectors, even if you don't think of it as a vector.  You have to control speed, angle of approach or take-off, air speed, wind speed, ground speed, etc.  So many things to think about.

So what are some of the math used by pilots when operating an airplane?  There is true airspeed, minimum landing distance, etc

 You can calculate your  true airspeed by any of the following:
1. taking half your altitude, adding it to your indicated airspeed
2. divide your airspeed by 1000 and multiply the result by 5.  Add this to your indicated airspeed.  
3. increase indicated airspeed by 2% per thousand feet of altitude.

The minimum runway length is the length of the actual landing distance/ 60 percent which results in a longer length.  Calculating the minimum runway length required for a wet landing strip is the same but you have to multiply the actual landing distance by 115% before dividing by 60%. 

If you want more detailed information on runway lengths check out this pdf as it has some very technical information, complete with the mathematics.

Temperature affects how much lift is needed to get a plane off the ground. It turns out the higher the temperature, the air density decreases so a plane needs more runway, a faster approach with a poor climb rate. 

With just a bit of looking, one can find all sorts of information on these topics and more.  It is also possible to find the same information in graphic form so students can learn to read graphs for a real world situation.

I just looked at three factors in detail and you can see the math involved in it.  This is cool.  I will look at more factors in future columns. 

As always, let me know what you think.

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Stuck in Transit.

Coming back from a conference, I landed in the local hub to a blistering wind storm with enough force to cause small plane wings to shake badly.  They were not flying so I ended up in a town where half the rooms were already booked for a conference.  I ended up at a place with no internet.  The weather cleared and I made it home in time for lunch.  It was back to work and no time off.  I discovered something neat about ratios and flying.  I'll share that on Monday.

Paper Circuits

zeros for a polynomials

Sunday, I gave a two hour workshop on using paper circuits in math at the local technology in education conference.

It was wonderful.  I had a room full of adults who were involved and having fun.  I assumed most people had never done paper circuits so I started at the basics.

First, I had them create a simple series circuit.  I gave them the supplies, a picture and let them do it, just the way I would in class.  I didn't even put the circuit drawing on the paper.  I made them do it themselves as part of the learning process.

The second step, was creating a parallel circuit with two rows of copper tape close to each other.  Participants had such a great time working to get the lights lit.  I showed a second way of creating a parallel circuit.  It was great watching them working together, speculating on why certain lights came on but then went out.  So much learning.

Over the next hour or hour and a half, I had them light two points on a line so they could eventually find the equation of the line.  They used parallel circuits and most everyone managed it.  I alternated working on projects with showing what could be done including creating your own switches.

Many of the elementary teacher in the group were weary at first but after I showed them a couple of things they could use with their students, they were much happier.  They were happy, they could use the lights to designate vertices, create pop-ups, and a few other activities.  I've never thought of using pop-ups in math but I can see ways they'd be good for elementary students.  

As time went on, a few began working on their own thing using techniques they'd learned.  One guy, successfully set up the circuits so when his boss proposed to his girlfriend, the ring would light up when the box was opened.

Another gentleman spent time working on a binary machine.  He has not finished it but promised to send me a short video when he got it up and running.  I inspired a few folks to just go off, play, and create.  Yeahhhhhhh.

Let me know if you've every used paper circuits in your class.  How did you use them.  Thanks ahead of time.

Student Based Learning

Over the length of the technology conference, there was one unifying cry.  How do we turn instruction from teacher centered to student centered.

I've been interested in making the change but I've never actually managed it.  I got some great ideas and suggestions so I know the steps I need to take.

One speaker gave some great ways to use google classroom so the students check in rather than you taking roll.  She spoke of ways to use comments to give immediate feedback.

In addition, she talked about creating assignments with few in depth instructions.  Let them use the internet to help find what they need.  She stated, students go through the stages of grief when the instructional method changes to one where they must take responsibility for their learning.

Another speaker had us look at teacher apps such as edupuzzle and discuss how we could have students use it to show their  learning. Four groups looked at four apps.  It was great because these are the pieces I've been missing to make the change.  When I get back, I am going to look at all the apps on the ipads, so I can see how the students can use them to show their learning.

It was suggested we assign fewer problems but require students to explain how they solved the problems by showing each and every step by using a video program. If they can explain the steps in detail, they know what they are doing.

Other ideas thrown out included dividing students into groups and each group will produce a 30 second to 1 min video discussing the theorems used to prove triangles are congruent, factoring quadratics, etc.  So many possibilities.

One of the keynote speakers suggested that social media and personal web pages for students is the new resume.  Their life is shown through Facebook, Tweeting, Instagram, and their own web pages.  If they apply for college, they can send potential recruiters so see who they are.  

I just opened my own Tweeting account.  I'm wondering if I can set it up so I can tweet reminders to them for tests, assignments, etc.  I don't know but its a thought.  This speaker gave me a reason to allow cell phones in the classroom. 

Perhaps my students will be more involved in class and more willing to learn by making a few changes in my classroom.  I have the ideas and now I just have to do it.  I'd love to hear from people who have made these type of changes.  What has worked well?  Let me know.  Please share.

Classroom Chef.

  While attending a technology conference, I ran across the classroom chef.  No, he's not someone you'll see competing on chopped, cake wars, or American's worst cooks. You will find him in the math classroom.

He has a menu page with appetizers, entrees, side dishes, desserts, and the take out menu.  Each type is composed of several different activities.  I checked out the page on triangle congruence where students created away to prove they knew all the postulates without necessarily being totally dry.

In case you don't know the name, you may have heard of his Mullet ratio study.  The Mullet ratio compares the part of the haircut that parties/the business part.  This is a great 5 to 6 day activity which he said got student participation.

He also has a whole collection of "Would you rather?" an activity which presents students with a choice of two things and the student has to justify their selection.  For instance "Would you rather 5 6cm pencils or 7 5cm pencils.  Explain your choice."

Those "Would you rather" will make some great warm-ups when I get back.  There is no right answer and it requires my students to justify their answers.  Many students need to practice explaining their choices.  Often mine will state they just guessed which is really what they did.

Another activity is one where they take a Desmos graph and play with numbers to see how these change the original creation.   I find it cool to play with the graphs. 

Yes, the author made a book and I'm going through it slowly.  The author goes with the idea of getting students to take over their learning rather than teachers doing the leading.  So far, the material in the book is great. 

Between this book and an all day session I attended yesterday, I feel like I am finally understanding how to create a more student focused learning.  I'll keep you posted.  Have a good day.

Videos and Micro Chunks.

I am currently in Anchorage, attending the state technology in education conference.  I have learned so much in the first couple of days.

One of the speakers shared some important information which was brand new to me.  According to the speaker, the longer a video is, the fewer people will watch it to the end.

Apparently, the magic number is 59 seconds.  If a video is over a minute, fewer people are willing to watch it compared to 59 or less seconds.

This speaker who is a math teacher, stated it is best to make very short videos breaking down the process to no more than 30 seconds long.  In fact, she suggested that you make a video for each stop when working any problem.  This makes it easier for students to rewatch the material. 

The longer the video, the harder it is for students to rewind to a specific spot where they got distracted.  In addition, by breaking it down to one video per step, a student only has to rewatch the one step they are stuck on.

The speakers comments support what I've read about breaking the steps down into micro chunks so students see everything.  We should not assume every student understands the jumps.  I am guilty of telling students they do not need to write down each and every step if they understand it.  Many hate to write things and will write down the minimum.

Others do not want to admit they do not understand the material.  I like the idea of creating small videos showing each step.  Students do not have to write every step but they can repeatedly watch the step until they get the hang of it.

Furthermore, micro chunking material into small videos encourages students to take responsibility for their learning.  The speaker also mentioned that when you make them do more for themselves, they tend to fight the shift, including watching videos, because everyone hates changing.

I think I'm going to have to find a bit of time to create these mini videos.  I'll let you know how it goes.  Let me know what you think.  Do you do it?  How does it work?

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Weighted Averages

  As part of my animation class, my students are learning about weighted averages.  Many of the students in the animation class struggle with math.  They don't all know their multiplication tables, they find doing regular problems but as part of the class, they have to understand weighted averages.

I started class by showing students how grades are calculated if the teacher has chosen to weighted averages.  I do and I know many high school teachers who do it.

Several students are athletes at school.  They know they must have a 2.0 or higher in order to travel but they had no idea how their grade point average was calculated.  They found it informative when i showed them.  I also explained how finding a college grade point average differed from the high school average.

These three examples are ones they are familiar with.  I took them outside of their area of knowledge to look at average balances or average sales per day in a month.  I brought up the idea that one of the students owned a business showing others how to set up an MMA fight in their town.  A few eyes brightened because two students are into MMA.

I talked about one of them creating a business where he teaches others to set up and run MMA fights in their location.  I made up some numbers like he earned $700 per day for 15 days, $1500 each day for 2 days and $800 per day for 13 days.  I lead them through calculating the daily average and pointed out the monthly income from this venture.

A few students looked stunned at the idea that creating a product they sold to others so others could do something on their own. This idea of creating a product to sell to others is beyond something they have ever thought of.  It is something that can be done from a remote village so they do not have to leave.

The village is pretty much accessible only by plane if they want to get to Anchorage.  I showed them the same math could be used to determine the average number of passengers carried per month or per year for the local airline.  Then I changed it to their bank account to find the daily average balance and daily average credit card balance using the weighted average. 

It was fantastic.  This was one of the first topics I had no trouble finding lots of examples that use weighted averages.  I loved it.

Now for a quick question to all my readers.  I am thinking of creating a series of lessons helping people create math videos using green screening techniques.  How many might be interested in this type of information?.  Please let me know.  Thank you.

Travel

  I am in transit right now.  I'm on the way in to a conference.  I know most of my students have no idea of the cost involved in travel.  This would be a great topic for one day.

Have the students plan a trip, figure out the cost for them to go to their dream destination, of the hotel, rental car, admissions to things, food, etc.

They could do this for one person and for their family.  The final cost could be quite shocking.

The trip from my village to the hub is $500 round trip.  From the hub to Anchorage is another $300 to $400 round trip, so about $800 to $900.  The hotel in the winter isn't too bad but if you have to calculate the trip using summer rates, it means a cheap hotel is over $100 a night.  I'm not going to talk about the upscale ones.

Everything is so much more expensive in the summer so costs about double.  The food and admissions stay the same but it can be rather expensive normally.  

A week trip for one person is probably in the $3000 range for airfare, hotel, car, food, etc.  If the family is composed of 6 to 9 people, that makes the cost jump. 

For me, I can fly 2 people in August from Fairbanks to Los Angles for under $1400 round trip which is just a bit more than here to Anchorage.

If students choose to look at the costs at different times of the year to see how much seasonal changes affect the costs.  It would be easy to create an excel spreadsheet with the costs so they could create graphs to compare the different costs.

Yeah, a great real life problem.  I've got to go catch my flight.  Have a good day.

Famous Ratios and Ratios Used With No Thought.

Yesterday, I let my mind wander.  It began with division before heading off to scales as in scale models, eventually settling on ratios.  There are several famous ratios we use in real life.  Mathematical things we don't even think about being ratios.

The most famous example is Pi.  Pi is defined as the ratio of the circumference to the radius of a circle.  How many times have you had students try to measure the circumference of a circle using a string that was later lined up against a ruler.  The ruler is also used to determine the radius.

This activity has lead to great  discussions on why their results are no where close to the actual number.  We've discussed stretch, lack of precise measurement, and all sorts of other issues.

The golden ratio is another famous one.  Basically it is the whole length/long part = long part/short part or approx. 1.618....  It appears in art, architecture,  geometry, and other areas.  This is even mentioned in Numb3rs and the Da Vinci Code.  It is said the golden ratio was used by the Egyptians while it was used by Da Vinci when he painted the Last Supper.  In addition, people who create labels for soda and such use it so the ratios look right on the bottles.

In addition, dentists use the golden ratio when fixing teeth.  Both Notre Dame and the Parthenon were built using the golden ratio.  There are more places you see it, in certain instruments or even with insects.  The golden ratio has a tremendous influence without our being aware of it. 

In the financial realm, there are many ratios used and the following are just two.
1. Price to earnings ratio which is used to determine if the price of a stock is reasonable.
2. Profit margins = Net income/sales.
I hadn't heard of these but if you are in business, you are likely to be quite familiar with them.

Even in maps, there are certain standard ratios found.  The USGS uses 11 different scales on their maps depending what the map is of.  If its of Puerto Rico, it will have a 1:20,000 ratio while the map of the United States is 1:1,000,000.  

Anytime you look at  road map, an atlas, or anything else that has a map, you are going to see a ratio which is often referred to as a map scale.  It might be 1 cm represents 20,000 cm or 1 inch represents 100 km.  It depends on how the scale is set up.

In addition, look at building plans whether for furniture or for houses.  They are all done to scale with a ratio such as 1:20, 1:50, or 1:100 in S.I. units or 1/4" or 1/8" for US units.  the 1/4" inch means 1/4" on the plan represents one foot when its built. 

All these ways and we don't give it a second thought we are using ratios.  We don't think about it at all.

Mistakes in Division

  Over the past few years, I've noticed a trend in my incoming students.  The majority of the students seem to make the same two mistakes when dividing.  I do not know where it comes from.  I have no idea where or when the misconception developed.

It is frustrating because I am not sure how or when to reteach the topic so students start doing it correctly.

First is when students divide, they do not place a zero in as a place holder.  An example would be dividing 5020/10.  So 10 goes into 50 five times.  The student writes 5 above and brings down the 2 but does not put a zero above it.  They bring down the 0 for 20 and put 2 above it because 10 x 2 = 20. 

Instead of 502, they come up with 52.  These students do not recognize they need a zero in there as a place holder.

I've been thinking of having students do their division either on graph paper so they can place one digit per column or use lined paper sideways.  If there is supposed to be a number in each column for the answer, it might provide an automatic reminder to put the zero in for a place holder.

The other situation is thinking the remainder is the number you place next to the decimal.  An example would be 13/5.  The student knows 5 goes into 13, twice.  They write 10 and subtract so 13 - 10 is 3.  They put 5.3 rather than 5.6 because 3/5 = .6.  It is something that occurs with great regularity. 

This one is a bit more challenging.   I am not sure how to have them think about converting the remainder into a decimal.  The only idea I have is to have them create a picture showing the remainder as a fraction of the original. Once they have a fraction, they can convert it from the illustration into a decimal. 

If anyone has any suggestions on ways to help students overcome these misconceptions, I would love to hear.  I realize I could just let them do the math on the calculator and not worry about these misconceptions but the first issue could translate into dividing rational expressions.  They might not put a zero when needed if dividing x^2 + x -3/x+1. In addition, they might not use the remainder properly.

Positive and Negative Numbers

  Last night, I thought about how we use positive and negative numbers in real life, yet we teach it using a horizontal number line.

We teach  children to count in a positive or negative  direction based on the numerical sign.  Even now when students stumble, I place a dot on the board and point in a direction, to prompt them.

I don't think we spend much time giving students context for applying these two concepts to real life.

We have mountains we can see from the porch of our school.  Its nice to connect positive numbers to climbing it since we are increasing elevation, and negative numbers to going down because elevation is decreasing.  The same could be applied to airplanes as the rise to their cruising altitude or descending to land on the ground. 

The stock market gives regular reports with positive and negative numbers as the Dow Jones shoots up or drops.  Daily temperatures rise or drop depending on the time of day. There are times when both the high and the low are negative numbers so the rising would be the positive part and dropping would be the negative.

Population growth can use positive numbers indicating it is growing or negative because the population looked at is declining.  Most people discuss weight as gaining or loosing it both number which fall into today's topic.  In addition credit card balances are shown with positive (items purchased) or negative (amount paid). 

The situation that hits most of us immediately is our bank account with deposits causing our accounts to increase, or withdrawals causing our accounts to decrease.  If we spend enough, we could owe the bank because we know have a negative balance.

Even general elevations such as Mount Whitney in California is over 14,000 feet above sea level while Death Valley is over 200 feet below sea level.  Then we have high and low tides, especially the Bay of Fundy which has tremendous differences.

Everyday, we hear words that indicate the use of positive and negative numbers such as "I'm short this month" meaning a negative amount compared to the amount needed.  I traveled 80 miles further than I anticipated today which is + 80 miles.

I think I'm going to use this topic as a welcome back to school activity where I have students brainstorm all the uses we have for positive and negative numbers.  There are so many situations my students could relate to.  Yeah.

Let me know what you think.

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Fractional or Rational Exponents

  The other day, I introduced my students to fractional or rational exponents.  First thing my contrary student asked "When is this really used?"  Off the top of my head, I stated you need it when you have different roots and you have to combine them.  It also makes it easier when differentiating in Calculus but other than that?  I don't know.

I worked before I became a teacher but the jobs I had did not require the use of exponents in any way so I had trouble answering his question.

It is used more than I realized and I've taught some of these uses without understanding I was using fractional exponents.

1. Financial industry.  They use these types of exponents when they calculate compound interest.  Fractional exponents are also used to calculate depreciation or calculating the increased value of a home.

Biologists use rational exponents to calculate surface areas of different animals as a way of comparing sizes using the formula S = km^2/3 where k is a constant and m represents the mass of the animal.

In music, the exact frequency is found using a rational exponent. The formula is f = 440 *2^1n/2 where n is the number of black and white keys above or below the 440 frequency.

Furthermore, the power generated for a ship uses rational exponents in the formula
P = (d^2/3 *s^3)/c.  d represents the ships displacement, s is the speed in knots and c is the Admiralty coefficient.

I saw it used in a problem comparing the surface area of spheres which I'd never thought of.

For all the apparent uses in real life, its very hard to find concrete examples for uses other than the few I was able to actually find.  I know for me, I see the use as being able to combine radicals of different roots with the same bases which indicates you are combining lengths to get a total.  Maybe in carpentry or trig? 

I would love to have others contribute examples so I can share them all with my students.  It is frustrating when you look and just cannot find that many

Thank you in advance.

Alligator Algebra

Algebra Alligators is a cute little free app designed to help students combine similar terms or practice multiplication or division of terms with variables. 

The idea is simple.  A problem appears on the screen and you select the correct answer from the list of possibilities.  If the answer is correct, the alligators get fed, if the selected answer is incorrect, the shot misses the alligator. 

Once the alligators are properly fed, they leave and you have fewer to feed.  This is a timed activity.  If you get killed because you did not feed enough alligators.

The screen to the right has four alligators.  The problem is at the top and all the answers are between you and the alligators.  The clock is at the top right while the four hearts indicate the number of lives you have available.

Note the problem is 3x + x so you would tap the 4x under the 2nd or 3rd alligator.

An alligator disappears when all the terms in front of an alligator are used.  This is from one game I played.  I've already answered a few problems.  I have 69 seconds left.

Usually you finish off one, then another but the problems seem to be pretty spread out across the solutions so it keeps you guessing as to which alligator will disappear first.

When you finish feeding the last alligator, you are congratulated for escaping the alligators in a certain amount of time. 

There are no levels but every time you replay the game you answer different problems so you cannot memorize the problems.

I enjoy playing this game myself.  I feel it is a good choice to use in the classroom to strengthen student skills.

Multiplication signs

  Yesterday, while I wrote about division, I realized many of my students arrive in high school missing some base knowledge for operations.

This lack of knowledge creates a major misunderstanding when students begin taking algebra or pre-algebra.

Too many students arrive in high school thinking multiplication is always represented by the "x".  So if you are multiplying numbers only such as 8 x 5, its is not bad but once we add variables into it like 2x + 3, confusion arises.

I've had student turn in papers with 2xx+3 meaning two times a number plus three.  I have to  teach them 2x means two times a number.

I've often wondered why we do not introduce the idea that multiplication can be expressed using a symbol other than x.  Why are we not using the * or dot to indicate it.  Many of my incoming students have trouble with 2(x + 3) because its not written as 2 x (x + 3).

I have not found any real explanation for why the x is used vs the dot in elementary.  I've read comments regarding using the x in elementary as it really doesn't matter.  I think it does or at least it does by the time you get to 5th grade.  I think its important to expose students to the different ways of showing multiplication before they arrive in high school.

Even in elementary, we can show students multiplication in at least three different ways, 3 x 5, 3 * 5, or 3(5) so they become used to using the other ways.  Unfortunately there is one problem with using the dot.  When working with vectors and scalars, you have a dot product which might confuse people but you don't usually get that until you are much higher in mathematics.

Is it practical to teach the three ways to multiply in elementary?  I believe it should since its almost like a progression and each method could be incorporated each year beginning in 3rd or 4th so by the 7th grade, students are familiar with the symbols and upper level teachers can focus on teaching other things. 

I don't know if this is possible but I do know it would make it easier to introduce new concepts in high school if I do not have to backtrack by teaching the various forms of multiplication signs.  Let me know what you think?

Illustrating Division of Fractions

  Things have changed since I first started teaching.  One of the big changes is the desire to have students understand the concepts behind the math.  When I was in school, you didn't learn about concepts, you learned to follow a prescribed process. 

You didn't vary from that process.  It was accepted you didn't really need to know what you were doing, you just needed to know enough to follow the steps to obtain the correct answer.

Over the years, I've discovered my students often did not understand or know the steps needed to solve a problem.

For instance, why do we change a division problem with fractions into a multiplication with its reciprocal?  After quite a lot of looking I finally found an explanation which makes sense.  When you divide one fraction by another such as 3/4 / 2/3  you multiply the top and bottom by 3/2 to get one in the denominator.

In all the years I've taught, this is the first time I've seen an explanation of why you multiply by the reciprocal.  When I got my teaching credentials, we used the "Flip the right one, not the wrong one."  Even at the local community college, we were encouraged to use that but never this other explanation.  I wish I'd known it then.
 
The explanation is so clear and easy to see but its not as clear when you try to draw a picture to show the same thing.  I honestly tried to figure it out myself.

 I drew the first block divided into quarters.  I divided the second block into thirds.  I had to divide each quarter into 3 more so the whole block consisted of 12 segments.  I then divided the 12 segments by 3 so I knew each 3rd equaled four segments from the first block.

So 3/4 equals 9 segments while 2/3 equals 8 segments.  3/4 / 2/3 is 9/8 or 1 1/8.  I am not going to tell you how many times I watched the Learn Zillions video before I finally got the hang of it.  I think I watched it close to 15 times. 

Yes I'm stretching my mind learning this but its cool.  We all need to learn something new every day.  Let me know what you think.  I look forward to hearing from you.

The equal sign

  Have you ever wondered if we are giving them a slight misconception when we teach equations?  As shown in the picture, these equations result in a number.  Most linear equations we teach also equal a specific number such as 2x + 7 = 15.  We have them solve it and they know the value of x is 4 but the idea the value can change becomes a bit more difficult for linear equations such as 2x + 7 = y because it does not equal a constant.

Most people see the end result of the equal sign when they go shopping.  Once they've checked out, the register totals all the prices and they pay.

How do we get students to go from looking for a set numerical answer to thinking about equality being a two way street.  I often teach it only one way such as 2x + 7 = 15.  I seldom even write equations as 15 = 2x + 7 or even as 2x+7 = 6x - 11.  This becomes a problem when my students take trig and have to work both ways when they are proving trigonometric equations.

According to one article I read, over 70 percent of middle school students use a running total when solving order of operations rather than understanding the whole process.  An example might be 4 + 2 + 8 = ___ + 5.  Many of these students will treat the blank as a place to write the sum for  the left hand portion of the equation.  They solve it this way 4 + 2 + 8 = 14 + 5, not understanding the left side equals 14 while the right side should equal 14 so the missing number should be 9.

The equal sign is a relatively new invention dating from the 1500's.  Before then, mathematicians used words such as yields, or gives.  Now the equal sign has several different meanings:

1.  It gives the result of a calculation such as 4 + 6 = 10

2. Decomposition such as 18 = 6 + 12

3. Equivalence between two expressions such as 3 + 9 = 2 x 6

4. The equal sign can indicate equivalence such as in 2/3 = 4/6

So it boils down to three different meanings for the = sign.  The result of a calculation, decomposition and equivalence.  Do we teach all three meanings or do we assume they understand the variations according to the context?

Let me know what you think.  I can tell you I need to work harder on conveying all three variations to help my students become more mathematically fluent.

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Warm-up

Visual Math in Architecture

Above is a beautiful picture of the Roman Collesium in Rome, Italy.  Look at all the beautiful geometric shapes of arches or semicircles on top of rectangles.  If you look down from above, its a beautiful ellipse or oval shaped. Imagine being able to calculate the equation for this specific ellipse.  With a bit of research, its possible to find the information so students can figure out the area of each of the arched entrances.  Students could possibly even determine the open area these entrances open up.

  Look at all the beautiful parabolas found on the golden gate bridge.  There are also nice lines and rectangles hidden in the bridge.  Image having students find the length of the bridge and the distance between the two posts with the beautiful parabolas so the students could create the equation for this piece of art.  They could also figure out the equation of the parabola at the end.

 

Look at the Arch of Triumph in Paris.  This is filled with some great mathematical shapes one, could find the equation for so many different shapes.  In addition, think about calculating the number of bricks used to build this, the equation for the arch, circles, and so many more things.  It just means you look at it and look for anything mathematical you can. 

This struck me as I'm putting together a short video on the Roman collesium and I realized its basic shape is elliptical so I can have students calculate the equation for it.  In addition, there are other shapes students can find the area for, equations, etc.  Look at public buildings, have fun annotating pictures and finding equations or areas. It is great.

Does Writing Help Improve Understanding?

  I have always been told if you can explain it in your own words, you show you understand the material, at least in English.  I wondered if the same thing applies to mathematics?

First of all, research indicates that improving a students ability to both read and write increases their learning.  In addition, writing can reveal misunderstandings, holes, and other issues with a person's understanding.

Most of us have never been trained to include writing in our math classroom.  I am the first to admit, I tried to have students journal their thoughts early on but I discovered they only copied their notes.  Since then, I've learned ways to sneak it in so they cannot just copy the notes.

Many students can solve any problem you give them but the minute you ask them how they got the answer, they tell you its done.  Others come up with the correct answer, only because they made an error in their calculations.  They don't care the process was not correct. I get told "I was close enough."  I try to explain close enough, especially in real life, could mean a bridge falling down or a building crumbling to the ground.

In addition, it may not be the actually writing that is important, instead it may be that writing requires students to think about their ideas and communicate those ideas.  When we require students to begin writing in the math class, we have to make sure to create a situation so students are willing to take a risk. 

I finally figured out how to implement more journaling in my class.  I give a warm-up at the beginning of the class period.  Since I collect these every week, I can have them write something on the back every day or every other day.  I do not include enough writing in my classroom but after reading two articles listed at the bottom, I have ideas.

Some of the suggestions include:
1. Have students explain in writing, how they solved a problem.

2.  Create solutions as if they are writing a textbook.  This would include the explanations associated with each step of finding the solution.

3. A short essay on what do you mean when you are asked to prove something.

4. Letters to the teacher to explain the student's confusion in regard to material taught.

5. Write a letter to someone who was sick that day to explain the material covered that day so the student does not get behind.

6. Create writing tasks which have students explaining their work complete with examples.

7. Free writing where students write for a short period of time on a set topic. 

8.  Teach students to apply the 5 W's (who, what, etc) when they write on a specific topic.

9. Have students work collaboratively to create a written explanation of their thinking.

10. Have students write a letter to help someone who is struggling with the material.  How would they explain the process and the concept.

This site has some very good detailed suggestions on using writing in the classroom.
Here is another one.   I have always been told if you can explain it in your own words, you show you understand the material, at least in English.  I wondered if the same thing applies to mathematics?

First of all, research indicates that improving a students ability to both read and write increases their learning.  In addition, writing can reveal misunderstandings, holes, and other issues with a person's understanding.

Most of us have never been trained to include writing in our math classroom.  I am the first to admit, I tried to have students journal their thoughts early on but I discovered they only copied their notes.  Since then, I've learned ways to sneak it in so they cannot just copy the notes.

Many students can solve any problem you give them but the minute you ask them how they got the answer, they tell you its done.  Others come up with the correct answer, only because they made an error in their calculations.  They don't care the process was not correct. I get told "I was close enough."  I try to explain close enough, especially in real life, could mean a bridge falling down or a building crumbling to the ground.

In addition, it may not be the actually writing that is important, instead it may be that writing requires students to think about their ideas and communicate those ideas.  When we require students to begin writing in the math class, we have to make sure to create a situation so students are willing to take a risk.

I finally figured out how to implement more journaling in my class.  I give a warm-up at the beginning of the class period.  Since I collect these every week, I can have them write something on the back every day or every other day.  I do not include enough writing in my classroom but after reading two articles listed at the bottom, I have ideas.

Some of the suggestions include:
1. Have students explain in writing, how they solved a problem.

2.  Create solutions as if they are writing a textbook.  This would include the explanations associated with each step of finding the solution.

3. A short essay on what do you mean when you are asked to prove something.

4. Letters to the teacher to explain the student's confusion in regard to material taught.

5. Write a letter to someone who was sick that day to explain the material covered that day so the student does not get behind.

6. Create writing tasks which have students explaining their work complete with examples.

7. Free writing where students write for a short period of time on a set topic.

8.  Teach students to apply the 5 W's (who, what, etc) when they write on a specific topic.

9. Have students work collaboratively to create a written explanation of their thinking.

10. Have students write a letter to help someone who is struggling with the material.  How would they explain the process and the concept.

This site has some very good detailed suggestions on using writing in the classroom.
Here is another one.

Encouraging Growth Mindset.

  It is February 1st.  I always surf the list of 80% off books on Amazon on the first day of the month to find books I want to buy.  Today, I found one I am truly interested in.  Its "The Growth Mindset Coach" by Annie Brock and Heather Hundley for $1.99.

No I haven't read it yet but I did buy it so I can download it to my tablet and spend time reading it.  It has a 4.5 star rating from 36 people.  Aside from the price, the big reason I down loaded it is simple.  It has both sample lessons and a month by month list.  Each month corresponds to a specific theme, complete with ideas, things to think about so you can use it all year long.

As far as I can tell it is a general book but it provides a starting point for those of use who need ideas.  This book is not specific to math since one author is an English teacher while the other is an elementary teacher.  I will get back to you on this book once I've read it.

In the meantime I found a lovely site called Growth Mindset Maths from England.  First thing I saw on the page was a great poster with a list of beliefs including one focusing on choosing challenging problems so we can keep learning and not giving up.  You can download the poster and place it around your classroom.

In addition, they provide some great sheets to help you help your students change their thinking.  For instance, the one on modeling a growth mindset has what to say instead of what is being said.  An example would be having the student ask themselves what they missed in the process rather than deciding they are stupid.  Or changing the "Its too hard" into "This is going to take some time."

There is also a wonderful presentation which helps students go through an activity which has them looking at ways to change thinking about things while they take notes to reinforce the material.  In addition, you'll find surveys and great material to help your students gain a growth mindset.

I plan to implement many of my suggestions in my classroom to try to eliminate the I can't or It's too hard attitudes. I have a few students who just cling to those beliefs so I need this to help me work on changing their mindset.

Let me know what you think.