Thursday, April 27, 2017

Previewing Material With Trailers.

Photo Manipulation, Alien, Foreign  I'm working my way through the book Teaching Like A Pirate.  I really enjoy the book because the ideas are conveyed through questions.  Questions phrased in such a way as to allow your mind to come up with solutions.

One question in there lead me to consider using movie trailers, like those you see on television which try to capture your interest.

After some playing around, I discovered iMovie allows a person to take some pre-made trailers and adjust them to fit your needs.  I made one for my Algebra II and one for my animation class.  This is actually a cool idea.

When I played them, the class sort of stopped and looked with confusion on their faces.  They asked me to replay the trailers so they could see what they missed.  They fell silent and began reading the words flashing across the screen.  The attention was awesome.

Here is the video:


iMovie allowed me to create this.  I hoope it uploaded properly but I wanted everyone to see the possibilities.  For the last couple weeks of school, I plan to do a unit on the math involved in getting man to the moon so this is what the trailer is about.

I took the retro trailer, added pictures, changed the wording and I had a cool trailer so students would be teased into thinking about the upcoming work.

Let me know what you think.

Wednesday, April 26, 2017

Slide Rule Followup

Slide Rule, Slider, ScaleIt was quite an adventure for both myself and the class.  Talk about confusion and frustration!  I've never used one and the video we used was great for the first couple of things but then we got totally lost. 

In case you missed the entry, my Algebra II class made slide rules this past Monday out of paper. I had them use tape over the paper to make it stiffer.  The implement worked much better this way.

Yesterday was the day we tried to learn to use them.  Multiplying by two was easy but it got harder when we tried 7 x 3.  We got sort of lost and when we tried 455 x 615, we were so lost, I'm not sure we could have found our way back to where we started.

One of my students complained it was too hard which meant she didn't understand what to do while another  got so frustrated, he threw it on the floor and slept.

Fortunately, there was a retired geologist visiting whose father taught him to use a slide rule so he was willing to help us.  It was great, once we understood we had to align the number one to the first number, count over the other number and read the answer.  It was illuminating because we could suddenly do it.

At the same time, we talked about trying to do serious calculations using this slide rule.  They shook their heads in awe because they couldn't perceive a world without the fancy calculators they use now. I pointed out the women who did the calculations that resulted in sending the man to the moon.

It is so mind blowing.  I read somewhere that the engineers had the women check the computer results because the computer was so new, they weren't sure about its accuracy.  Next week, we'll watch the movie Hidden Figures to learn more about this. 

I'm hoping the movie will show one student what it all has to do with math.  I enjoyed learning to use a slide rule. Its not something I"ll use regularly but I can teach it again next year in conjunction with the movie and showing how important math is.

Let me what you think.  I love learning new skills and this is one, I'll have to practice.  Have a good day.

Tuesday, April 25, 2017


Free stock photo of black-and-white, cartoon, donald duck, spotlight

As you know, I teach the math of animation as one of my classes in high school.  I have been integrating Pixar in a Box from Khan Academy with actual math work.  Since this is a short week, I'm taking a bit of a break from teaching to let students create a short animation in the old fashioned way before computers.

I"m starting with a You Tube Video showing how Walt Disney people created the early animated films one slide at a time.  There is math involved in these early films.  Some of these early films used over 50,000 individual drawings. In addition, the pages have to be presented at a specific rate so the finished product looks great.

The going rate for most feature films was 24 frames per second and animated movies were created using this same rate. Now a days the rate is 30 frames per second to meet NTSC standards.  This means it takes 1800 frames to create one minute of animated film.  A five minute long film requires about 9000 frames. 

The original process required the best artists to make the main drawings with gaps in between so another person called the "inbetweener" could fill in the gaps making the action much smoother.  These first drawings were only of the characters, the background was added later.

The original process required people to plan the story ahead of time so they knew what was going to happen when.  This allowed the main artist to create the key action points so the inbetweener could provide the connecting frames.

I am going to have my students create a few seconds of film with the beginning of a story.  I plan to use my iPad to actually film each drawing so as to create the actual film.  I'd like to show these creations at the next assembly or at graduation. We'll see where the principal will let me do it.

Again, I'd love to hear your thoughts.  Thanks for reading.

Monday, April 24, 2017

Old Fashioned Technology

File:Vintage Small Slide Rule, 4.75 Inches in Length, Made in England (9610232930).jpg
By Joe Haupt from USA

I've been working my way through the book Teach Like A Pirate by Dave Burgess.  It had so many questions leading to ideas that I've taken a few for use in my classroom.  My Algebra II class has been working with exponential functions, logs and natural logs recently so I thought we'd do something different.

For the warm-up, I sent the students to a web site which gave information on the real ladies featured in the movie "Hidden Figures" to give them a bit of background on how people worked equations before hand held calculators or computers.  I find it fascinating the engineers had the women check the computer calculations prior to the launch.

A short discussion followed this on how they ran the calculations use a slide rule instead.  Of course, I found a template on the internet so they could make their own.  After having them make it, I played a video to show them how to use the slide rule.

Did you know there is a slide rule museum with a virtual slide rule and instructions for doing math on it?  There is.  So I sent them there to read the instructions to figure out how to do more.  And its being finished off using the slide rule to actually solve a few problems.

You might wonder why I'm doing this?  Well, it gives them a bit of a history of math in terms of calculating machines used to send man to the moon, provides the history of real people who used it, and it gives them a lesson in the use of older technology.  No I don't know how to use a slide rule but I'll be learning along side them. 

When I got to college, calculators were used by almost everyone in class.  These were HP's and TI's but you could do everything on them, so who needed a slide rule.  I wanted to change up the routine a bit because students only have two days of instruction this week.

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

Thursday, April 20, 2017

Taking Movie Notes

Film Reel, Cinema, Film, Movie, Reel In the last week or so, I've been showing videos on class but students only wanted to watch. They didn't want to take notes.  They do not get as much out of the videos as they could.  I realized, no one had ever shown them how to take notes.

 Although about half to three quarters of my students speak English, they are classified as English Language Learning.  This means they may not fully understand everything they hear on a video.

I think previous teachers never considered math as having its own language and assumed since they speak English, they understand language well enough to take notes.

I don't always have time to sit down and preview the movie before showing it so I can't type up a list of questions.  Instead, I'm showing the movies in class but taking time to stop it so students can write down the important information.

Students at this school, arrive in high school without certain skills including identifying the important information, even on a math video.  They also do not know how to read a math textbook to identify important information.

Two skills which are important in today's world.  When I got to high school, I didn't quite have the note taking skills down but I at least wrote everything I could because I was afraid I would miss something.  It wasn't until I started teaching, I learned how to distinguish among important vs unimportant information.

My plan is to continue showing videos but slowly move from me being in charge of stopping it, to letting students signal when the video should be stopped so they can take the note.  Eventually, I want to post the videos with questions integrated so they can watch and learn after school or if they are absent or traveling.

I don't think its just my students.  I think taking notes while watching movies has slipped by us as teachers.  We don't think to take the time to make sure students know how to take notes or use the video in an effective way to increase learning. I have in the past assumed a student who was watching the video would remember what they saw but I'm not sure that is correct.

Check back tomorrow, for suggestions on using videos effectively.

Wednesday, April 19, 2017

Reading and Creating Math Problems.

Newspaper, Paper, Pencil, Glasses  When you work in a school where students are either English Language Learners, or students who live in a certain amount of isolation.   They do not always relate to the context of word problems such as the ones dealing with trains.  There is one big train route in Alaska but its nowhere near here. 

I've rewritten problems to use local names and snow machines instead of cars but to encourage reading, I've gone to various news outlets.

I look at yahoo news,  major newspapers, and online news to find articles which might catch student interest.  I read the article and design mathematical questions which use the material.

One time I found an article about a teenager who ended up owing $35,000 in texting costs because she had too much fun texting and went over her data amount.  This story took place a few years ago.  I had the students figure out the cost per text since they had total cost and number of texts.  In addition, I asked them to calculate the cost of texts if the plan had had unlimited texts. We used a cost from the local cell phone provider.

Another article was one of the overcharging taxi's.  The ones who charge way more than the going rate to visitors.  I used one to calculate the percentage of overcharging for the taxi based on the normal cost of the trip.

One of the best articles I've ever used talked about the amount of tax the city of New York lost in a month due to people buying cigarettes elsewhere and selling them below cost illegally.  This was wonderful because many of my students are smokers.  They had to calculate the amount lost per pack, amount lost per month, and per year.  It turned out to be in the millions.  The students were shocked at the amount of tax New York City lost every year.

One day, I found an article on a dog who resembled a lion was sold for a million dollars in China.  Rather than writing the questions myself, I had students come up with several mathematical problems which could be answered using the information in the article.  They came up with questions like "What was the cost per pound of dog?" , "What was the cost per cubit ounce of the dog?" and others.

Yahoo News has some really great problems in their odd news section.  If you do a web search for "odd" or "strange" news, you'll find lots of places to look. Give it a shot.  Make your problems more interesting while incorporating reading.

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

Tuesday, April 18, 2017

Spacing Practice

Elephant, Proboscis, Shield, Memory

One way to increase retention of learning is to space practice sessions for old material.  Most teachers do not worry about checking students to see how much they remember because they have so much to cover throughout the year and usually do not have enough time.

Teachers struggle to do everything required while trying to teach between state mandated tests, school mandated assessments (AIMS, MAP, etc),  holidays, student trips, and other interruptions.  What are some ways to help students retain knowledge while working around interruptions?

There are ways to created spaced practices even with all the interruptions.  Some things teachers can do are:

1. Study the information more than once.  This means, it can be referred to when teaching other material if the information is applicable.  For instance, you can review rules of exponents when teaching exponential functions and logs because the rules are used by both. 

2. Regular testing such as using daily quizzes, weekly quizzes, exit questions, or warm-ups. This should be low stakes rather than waiting for the main test.  Ask questions to see if they remember yesterday's material, last week, or last month.  It is the act of retrieval which increases retention.

3. If you want to remember it, it must be recalled from your memory.  Ask a question about something already learned.

4. Keep returning to content considered important.

5. Do not do the practice, practice, practice because it does not help student retention. 

6. Retrieval is best when a person has already had some forgetfulness occurring.

7. It is easy to think you are getting better when in reality gains can quickly disappear.

Make retrieval practice a part of your daily routine.  If you taught the rules of exponents yesterday, ask questions on them today, in three days, in a week, in two weeks, in a month and the student is more likely to  remember the material.

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


Monday, April 17, 2017

Memory Vs Thinking

Puzzle, Learn, Arrangement, Components Today's thoughts came out of wondering if spiral learning and interleaving were the same.  I stumbled across an article on spacing and interleaving from 2014 with a quote that made me sit up with a whoa.

The quote "We rely on memory rather than thinking" from Hattie and Yates, Visible Learning and the science of how we learn.

This quote perfectly identifies the overwhelming thought of teaching as it has been for so many years and seems to be continuing even now.

Why else would we teach students to follow processes when solving problems rather than looking for connections.  I wonder if this is why so many of our students are unable to transfer knowledge from one situation to another or why they barely remember yesterday's lesson.

When teaching students to solve word problems, we often tell them to look for key words which may nor may not provide the correct information, make a plan, etc.  Most of my student just look at the numbers, hoping to choose the correct operation.  They don't stop to think about what the problems is asking or how they might solve it.

I know most teachers do not even ask questions designed to have students think.  Usually the questions are "What is the first step?", or "How do we get rid of the constant?".  Questions designed to ask about processes.  In addition, most students do not know how to ask each other questions to open a discourse.  Most students ask "Do you know how to do this?".  If the answer is no, they quit and if the answer is yes, they want the other student to let them copy.

I found this list of 100 questions which can be used to move past the standard few used in the classroom.  I like some of the questions such as "How did you reach that conclusion?" or "What is this problem about, can you tell me?  Questions designed to make students think about their work.

Note, I am not saying Memory is unimportant.  It is but too often students use the processes they've learned rather than thinking about the problem itself.  I'm saying as a teacher, we need to encourage thinking through every facet of problem solving.  Don't tell, ask questions to invoke thinking.  Ask for explanations rather than steps.  Ask why.  Ask them to tell you.  Ask, Ask, Ask.

Perhaps by encouraging more thinking, we can move past simple memorization, into real understanding of mathematics.

Friday, April 14, 2017

Relating topics

Logarithm, Board, Mathematics, Pay  Since reading about the big idea, I've seen more and more ways I can talk about concepts throughout the classes. 

I've been able to discuss solving equations with my Algebra II class today.  They are solving exponential equations with same bases.  Since the exponent were often algebraic equations, students had to use the same steps as they had when solving simple algebraic equations.

We are also discussed the rules for using powers because they applied in this situation. When students rewrote the bases so both sides had the same base, they had to apply the power to a power rule.  It was so great being about to show this. 

Next week, they will be learning about logs which follow the rules for working with exponents such as when the bases are the same, the exponents are added.  Now there is a second area where the rules of exponents apply other than the section on exponents.

In algebra I, it is great talking about using the same steps to rewrite literal equations as are used for finding a numerical answer from an algebraic equation.  It is lovely discussing a situation where the same steps apply but you get a variable rather than a numerical answer. 

Several of the teachers use the lattice method for multiplying multi digit numbers together.  I've taken the same lattice and used it for multiplying binomials and polynomials together.  Many of the students learned it in one class period because they were familiar with the basic process.  The same can be said for addition and subtraction of polynomials.  

I use diagrams showing multiplication of two digit numbers such as 53 times 22 using 10 block and ones to make a square with the tens units squared for hundreds etc.  I use the same diagrams to show binomial multiplication so they can see the x^2 is similar to the hundreds or 10^2.  It shows multiplication works the same for both.

I am going to spend some time over the summer looking at redoing the order I teach concepts so maybe I can put ones together which share common concepts.

I'd love to hear what you think.  Thanks for reading.

Thursday, April 13, 2017

Followup on Reading Standards.

Still Life, Teddy, White, Read, Book  Yesterday, I tried applying reading standards to solving word problems and it went so well.  I started by selecting several word problems that are slightly below their reading and ability level so they could do the problems.

I began by reviewing these terms:
1. Main character
2. Setting
3. Event or what was happening
4. Inferences which need to be made
5. Summary - a sentence or two to summarize all the information and the answer.

The second step was showing a problem on my Smart Board so I could model my expectations.  The problem was something like:

"There were 6846 seeds in the bird feeder.  6 bluebirds and one baby bear decided to split the seeds evenly among themselves.  How many did they each get?"

We decided the main characters were the 6 bluebirds and one baby bear.
The setting: its happening by the bird feeder.
The event: splitting the seeds up evenly.
Inferences:  6 birds plus one bear means 7 totally.  Splitting means division.
I had the students do the math so we had an answer for the problem

The summary - The 7 animals split the 6846 seeds evenly among themselves so they had 978 seeds each. 

All my students had a great time applying their reading skills to solving math problems.  It showed the skills they use when reading a book can also be used in mathematics.  Most of my students prefer using these skills than the usual ones associated with problem solving in math.  We will do this again next week with more complex problems.

Let me know what you think.  The English teacher loves this idea because it answers the question of will they ever use it outside of the English class.  Thank yo for reading.

Wednesday, April 12, 2017

The Big Idea

Light Bulb, Current, Light, Glow  I come across interesting material when I checking out a variety of topics on teaching the big mathematical idea because it is easier for students to relate to connected concepts.

If you think you know what they mean by big idea, you might be surprised by its definition.  I was.  Think of the big mathematical idea in this way.

The big mathematical idea is a statement or idea central to learning mathematics.  It links several mathematical concepts into a coherent whole.  It turns out there are several components to the big idea.

1. It is a statement.
2. It is an idea that is central to learning mathematics.
3. It links everything together so it makes sense.

For instance, if you stated that all numbers, fractions, decimals, numerical expressions and algebraic equations can be represented in so many different ways that have the same value.  This is a big idea because you can find equivalents which have the same value.  2 + 4 has the same value as 5 + 1, or  one more  than 2.5 + 2.5.

Big ideas apply across all strands of mathematics rather than applying to only one strand.  Ideas such as "Geometric figures have attributes that can be classified and described" or "Numbers represent sets of items that can be composed or decomposed."  Furthermore, these ideas can be applied across grade levels which makes it more vertically aligned.

Each big mathematical idea connects learning together under an umbrella.  They build a better foundation because the connectivity in inherent in the ideas which leads to deeper understanding.  Yesterday my Algebra I class started rewriting literal equations for one term.  One of the students noted they were solving them just like they did their algebraic equations.  He saw the connection.

Using big ideas helps students become more flexible, generalize knowledge, improve problem solving, makes mastering new facts and procedures while and allows for better transfer.

I am from a generation who learned math by focusing on processes but for me trying to figure out the big idea is difficult.  I did find some help in this presentation for the big ideas in math.  Although the presentation is for elementary school, it gives me a starting point so I have ideas to use as umbrellas in class next year.

Let me know what you think.  I always love hearing from people.

Tuesday, April 11, 2017

Process vs Understanding.

Mathematics, Formula, Physics, School  While looking up something on the internet, I came across a blog written by a young person who recommended students focus on learning the steps involved in solving problems.

He said a person should carefully look at the steps and determine the exact number of steps needed to solve a particular type of problem.

He went on to tell people to use those exact same steps when solving the same type of problem.  Practice using those same steps and eventually, you'll become a whiz at solving problems.

Toward the end of the blog, the author admitted that he did well in calculus even though didn't always understand what he was solving or why it worked.  He was proud of the grade.  I mention this because it supports the need for students to understand what they are solving and why it works.

His advice means a student is learning the mechanics of solving problems without having to understand anything more.  Unfortunately, we are expecting students to know when to use certain problems or why they are applied in certain situations so his advice may not work as well today as it did when it was originally written.

According to a page on Understanding Mathematics at the University of Utah, you understand math if you:
a.)  are able to explain mathematical concepts using simpler concepts and facts.
b.) make logical connections between different facts and concepts.
c.) see a connection between something new and something in math you know.
d.) can identify the principal of any math you do.

It is suggested that if you miss a point in the process, get clarification immediately because math builds on previous concepts.  Its like building a house with one joist missing.  Your house may begin sagging and eventually fall.

Memorization only takes you so far but developing an understanding leads to so much more.  So the author of the blog provides a good reason for learning to understand mathematics.  Let me know what you think.

Monday, April 10, 2017

Explaining Thinking

Smartphone, Handheld, Cell Phone  We all know students have access to apps and programs on the internet which will do the work for them.  They put the problem in and the answer pops out.  If they choose the right app or web site, the solution will include all the steps necessary to "show their work".

They get great scores on the daily work or home work but when the test arrives, they are totally clueless.  This is something that will not change unless mobile devices and the internet disappear.  So what do we do to make sure our students are learning the material.

I have seen several suggestions to help students learn the material while using apps to solve problems for them.  Suggestions which fall under the heading of explaining their thinking.

1.  Post two different worked out problems which are almost the same but just a bit different.  Have students discuss the similarities and differences.  This could be done either verbally or via collaborative technology.

2.  Post a problem from the new material with all the steps. Have students explain what is happening for each step.  If they understand the material from the previous section, they can build on prior knowledge.

3.  Post a fully worked out problem and have students write in what is happening with each step, much like the the examples in most text books.

The idea behind having students write out the explanations for each step is actually two fold:

First, it is a way to access student understanding of the material.  If they cannot write explanations of the steps, it means differentiation and scaffolding may need to be implemented.

Second, it is a great way to monitor student thinking and identifying misconceptions.

As teachers we've reached a point where students need to know more than the mechanical steps used to solve a problem.  They are required to understand the mathematical reasoning behind solving problems and they have to know how to explain their own thinking.  This is a complete paradigm shift from even 10 years ago.

So to ensure this, we must change our expectations and our methods to help students learn to explain their thinking.  I'd love to hear your thoughts on this.

Friday, April 7, 2017

Relating Reading to Math

Teacher, Theory Of Relativity, Equation When I was growing up, math and English were considered two different topics with absolutely no relationship.  Teachers did not try to connect the two.  We didn't have to explain anything, just get the write answer for the problem.

Even recently, when our students are expected to explain their thinking, it is hard to know how to teach students to do this when often teachers do not know how to model their own thinking.

I came a cross a great presentation  in which the person actually relates student learning from English to math.  It shows how these terms apply in math.

In English students are expected to identify the main characters and setting, find the main idea, implied main idea, and supporting details.  They need to make inferences, draw conclusions, and summarize.  If I told you math uses the same skills, most people would laugh their heads off and disagree.  My students would try to commit me to the local loony bin but I assure you, it can be done.

Word Problems in math contain all the information so you can do the same things:

Example 1.  Peter took Jen to the movie theater to see the latest Star Wars movie in 3D. Tickets cost $14.00 each.  How much change did Peter receive if he paid with a $50 Bill.

The characters are Peter and Jen.  The setting is a movie theater.

Example 2. John is replacing tiles in his entryway with 1 foot by 1 foot white tiles.  If his entry way is 6 feet by 8 feet, how many tiles will he need?

The main idea he is having to tile an area so this is an area problem.

Example 3. Joe and four friends went to the movies last Friday night.  Each ticket was $7.50 each, how much did they pay for all the tickets.

This reference uses inference because they have to infer that Joe is included in buying tickets so 5 tickets are purchased, not 4.

This is cool.  It is the first time I have ever seen someone who connected reading standards to word problems in math.  It adds a new dimension to interpreting word problems.

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


Thursday, April 6, 2017


Sports, Car, Racing Car, Roadster  No R.A.C.E. is not talking about the Indie 500 or any other car race. It is a process designed to help students create answers to a constructed response question.

Yesterday, one of the teachers gave us an overview of the process.  As a math teacher, my first impulse is simply, "My kids don't need to know that" but I realized they do need it, so I thought I would share it with readers.

R.A.C.E, is an acronym for Restate, Answer, Cite & Explain although in math, the C could mean compute and check.

Restate is to reword the question into a statement which becomes part of the answer.  Answer is the next part where the student provides the answer to the question.  Cite is where the person sites the evidence to support the answer but in math, the student would provide the numbers used to calculate the answer.  Explain might be showing their work or explaining the material read in a book.

What the students have when they are finished is a well written detailed paragraph containing the answer and their thinking.  It changes explanations from I multiplied two numbers to I had to multiply the length - 6 and the width - 4 to get the area of the rectangle.  Much more specific, detailed information.

The two places in math I can see using this structure is when students read the textbook or for word problems.  Since I work mostly with English Language Learners, I have to provide structures and scaffolding.  This process is great for my situation.

According to several things I read, the restate and answer are the same but the cite is the place they show their work and explain is discussing how they solved the problem or the strategy used. So a three step view of this is:
1.  Students must answer the question.
2. Students need to explain their answer.
3.  Students need to prove their answer is correct.

One idea to practice the process is to create "Entry tickets" which require students to create a short constructed math response to a question and each has a rubric at the bottom reminding students what is expected.  The question might be like Jo and June both worked the problem "96➗ 4 + 2.  Jo got 48 while June got 12.  Which student is correct.  Explain your answer.  These should not take more than 10 minutes.  This is something which could easily be used as a warm-up or bell ringer. 

I just found something that talks about math problems and reading in a way that shows a relationship I've never seen before.  I'll be sharing it with everyone tomorrow.  Let me know what you think about this. 

Have a good day.

Wednesday, April 5, 2017

The Math of Mixing Car Paint.

Painter, Paint Cans, Brush, Paintbrush

If you ever had your car repainted or needed touching up, the painter has had to mix paint for the car.  According to a chart produced by Sherwin-Williams automotive finishes, most paint follow certain ratios of thinner to paint. 

Mixing ratios are often given in parts or percentages.  If given in parts, the basic measurement has to be used consistently throughout the mixing process.  For instance, if using a 4:2:1 ratio, it means 4 parts of the base product, 2 parts thinner and 1 part hardener.  It might be 4cups, 2 cups, and 1 cup or 4 tbsp, 2tbsp, and 1 tbsp. 

If its expressed in percentages such as 25%, the percentage needs to be changed to a fractional equivalent.  25% is 1/4 which is the ratio needed to mix paint.  1/4 means 1 part thinner or reducer to 4 parts paint base.

In addition, the type of paint often determines the ratios used.  For instance, if it is a single stage paint which is glossy and durable and you only need to put on a few layers to be done.  The usual ratio is 8/1/1, or 8 parts paint, 1 part reducer/thinner, and 1 part hardener. 

On the other hand if you use a two stage paint which starts with a base coat paint with a ratio of  1 to 1 or 1 part paint to 1 part reducer.  For a clear coat, the ratio is either 4/1 or 2/1 with say 4 parts clear coat to one part hardener.

Furthermore, the type of paint used will determine if you only mix paint and thinner, or whether hardener is needed.  For instance, the solvent based metal paints is thinned down using a 2 to 1 ratio of paint to thinner.  No hardener is needed  while the two pack acrylic requires a 2:1:10  the 10 being a 10 percent mix of hardener.

So much real math using ratios in a real life setting.  It wouldn't be hard to create an exercise requiring students to plan to repaint their own car. 

Let me know what you think.

Tuesday, April 4, 2017

Building Metacognition in Math.

Turn Pen, Manga, Anime, Digital Design  Its well known we need to build metacognition in math to help students learn the material better.  Often students do not think well of math due to the idea that mastering math in the classroom is learning a bunch of formulas, rather than understanding math can be meaningful.

 Metacognition can help increase the meaningfulness of instruction so math makes more sense.  We know students can do the calculational part of the problem but fall short on applying meaning to the problem.

If a student is given a problem where they have to determine the number of buses to order for a field trip where they have the number of people going on the trip and the number of passengers in each bus, students will carry out the algorithm correctly.  Their downfall comes when they have a remainder.  They often give the number of buses with a remainder, or they do not round up to include the remainder.

There are three ways to look at metacognition in math.

1.  Beliefs and intuition - This is one based on the ideas people bring to the classroom and how do they shape the way we do math? One big belief which has developed is that the classroom math is just formulas that do not relate to the real world.  Students find math boring because of this.

2. Knowledge about your own thought processes. - It is important for students to know how they think because problem solving requires you use what you know efficiently. Student understanding of a task and their ability to solve it are effected by what they think they can learn.

3. Self awareness or self regulation - How well does a student keep track of what they are doing when they solve problems? How well do they use observations to guide their problem solving? In other words students need to develop an awareness of their thinking and their progress as they are solving problems.

As teachers we need to model our thinking for students so they learn how to think about thinking.  It is recommended we use the words "I think,,," when attempting to show students your thinking and so they see the process.

Providing graphic organizers also help create a visual representation of that thinking.  Graphic organizers help the mind focus on the important parts of our thinking and we can look at it when we are done.

Finally, teachers should plan the behaviors they are trying to model by creating a script or a plan ahead of time which includes possible misconceptions so they can be addressed a head of time.  One misconception my students have is they do not need a zero place holder when dividing.  They always leave it out.

Let me know what you think about this.  I love hearing from others.


Monday, April 3, 2017

Pens and Math

Pen, Fountain Pen, Ink, Gold, Writing  You are probably wondering why I'm writing about expensive pens in a math column?  Believe it or not, these pens are interesting from a mathematical point of view.

The other thing about these three pens is they are not your standard pens.  They are fountain pens which have 18 carat nibs and can be refilled with ink but at these prices, I don't think I'd use them.

The first pen was built using a practical application of math in its creation.  One, we don't usually think about. 

This pen, Fulgor Nocturnus Pen created by Tibaldi of Florence Italy, sold for $8 million dollars back in 2010 at an action in China.

So what made it capture my attention?  It was created using based on the Divine Proportions of Phi also known as the golden ratio or 1.618.  The ratio between the cap and the visible barrel with the closed cap is a perfect 1.618.  The pen is decorated with 925 black diamonds and 123 rubies.

Several other expensive pens in the $200,000 to $500,000 range are not your usually cylindrical shape.  They have hexagonal or 6 sided barrels.  One is the Gotica pen with a beautifully decorated barrel encrusted with 892 diamonds, 72 rubies, and 72 rubies costing just over $400,000, while another is the La Modinista pen with 5,072 diamonds and 96 rubies on the hexagonal barrel for only $250,000.

It might be interesting to have students design a pen using the golden ratio just as the Fulgor Nocturnus.

Let me know what you think.  I found this rather intriguing that pens can be created using the golden ratio.

Friday, March 31, 2017

Google Streetview/Google Maps in Math

Map, Location, Navigation, Symbol 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.

Harmonica, Music, Instrument, In C Major  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

Geese, Birds, Goose, Domestic, Fowls  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

Lightning, Storm, Night, Firebird  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.

Banner, Header, Mathematics, Formula  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.

Cogs, Gears, Machine, Mechanical  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

Relay Race, Competition, Stadium, Sport  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

Logo, Globe, Time, Time Of, World Time

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.

Search, Math, X, Unknown 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.

Plastic, Polymer, Granules  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


Geometry, Mathematics, Cube, Hexahedron  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


Tux, Animal, Bird, Book, Books, BookwormSorry, 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!

Computer, Macbook, Tablet, Editing  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

Update on Google Classroom.

Tablet, Books, Education, Desk  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

Database, Storage, Data Storage 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.