If you are old enough, there was always someone in the family who built a huge 4 by 8 railroad using models. I remember seeing one in the family with a town, forest, bridge and a lake as part of the journey.

If the builder did things correctly, everything looked perfect because the scales remained the same for everything. If not, the trees might be a bit too small and the lake more like an ocean.

Did you know the scales for model railroads were not always the same? I didn't until I researched this topic. The National Model Railroad Association has some excellent information if you want to read about this topic in more detail.

In model railroading, scale means it is a smaller version of the real thing. O scale is also known as 1/4th inch scale which is a ratio of 1 to 48 while the HO scale is 1/8th inch scale with a ratio of 1 to 86 or 1 to 87. The N scale made its appearance in the 1970's and has a ratio of 1 to 160 while the Z scale has a ratio of 1 to 220. This means the O scale is the largest while the Z scale is the smallest.

Another one, the S scale as defined in 1943 is 3/16th inch or a scale of 1 to 64. At one point this scale almost disappeared but has undergone a Renaissance.

There are a few others but these are the main ones. Now to add to the confusion, one has to keep in mind, one has to keep in mind the gauge or distance between the rails. For instance, the distance between the rails for the O scale is a bit more complex because it normally requires 3 rails instead of two and the distance represented is 5 feet.

Its all a bit confusing if you are not familiar with this topic because the distance between the rails must also be done properly so everything looks proper. So what do you do with this information? It makes a good project involving research and drawings which are properly scaled. As part of the research, students can determine which scale is used with which manufacturers or the train such as American Flyer or Lionel.

I have been traveling and this is being written in an airport between Fairbanks, AK and Burbank, CA. I'm at a conference till Monday when I head home.

## Friday, July 28, 2017

## Thursday, July 27, 2017

### Creating Escher Tessellations in Class

M.C. Esher born 1898 and died in 1972. He is best known for his use of tessellations in his art work. You've seen his work even if you don't know it. Among other things, he did several drawings where stairs seem to go back upon itself.

He is perfect for using in a math class during tessellations and transformations because he uses both in his artwork. This YouTube video does a lovely job of introducing M.C. Escher's work while discussing the mathematics of it.

The question becomes how did he create it? Well there is a wonderful 32 page teachers guide from the Akron Art museum on it.

The guide includes the mathematics behind the topic. After introducing M.C. Escher, the first lesson begins by discussing different types of symmetry. Next students are expected to identify the type of symmetry used in the examples.

Lesson 2 focuses on three dimensional shapes such as those found in crystals. The lesson includes information on faces, vertices, and edges so they get the vocabulary. Lesson 3 goes on to look at architecture, parallel and perpendicular lines, and creating cubes out of isometric paper. Lesson 4's topic is reasoning through the use of a Mobius strip and includes real life uses. All four lessons focus on the mathematics involved. The remaining part of the pdf focuses on art. I like the way the math lessons are created.

You Tube has several nice videos on creating general tessellations in the Escher method including this 11 minute one which is slow and takes things step by step. At the end students have one but the form is not specific. If students want to create a more specific shaped tessellations check this video shows how to create a Escher Bird Tessellation step by step on sketchpad.

In addition Tessellations.org has some great instructions using tracing paper and equilateral triangles. In fact the whole site is devoted to tessellations. They even have a version of Angry Birds in a tessellation form.

I'm finishing this up for two reasons. First, there have been two power outages already this afternoon in the last 15 minutes and second, I have to finish packing for a trip I am taking to Los Angeles. I needed to get ahead so I do not miss anything.

Let me now what you think. I love hearing from people.

He is perfect for using in a math class during tessellations and transformations because he uses both in his artwork. This YouTube video does a lovely job of introducing M.C. Escher's work while discussing the mathematics of it.

The question becomes how did he create it? Well there is a wonderful 32 page teachers guide from the Akron Art museum on it.

The guide includes the mathematics behind the topic. After introducing M.C. Escher, the first lesson begins by discussing different types of symmetry. Next students are expected to identify the type of symmetry used in the examples.

Lesson 2 focuses on three dimensional shapes such as those found in crystals. The lesson includes information on faces, vertices, and edges so they get the vocabulary. Lesson 3 goes on to look at architecture, parallel and perpendicular lines, and creating cubes out of isometric paper. Lesson 4's topic is reasoning through the use of a Mobius strip and includes real life uses. All four lessons focus on the mathematics involved. The remaining part of the pdf focuses on art. I like the way the math lessons are created.

You Tube has several nice videos on creating general tessellations in the Escher method including this 11 minute one which is slow and takes things step by step. At the end students have one but the form is not specific. If students want to create a more specific shaped tessellations check this video shows how to create a Escher Bird Tessellation step by step on sketchpad.

In addition Tessellations.org has some great instructions using tracing paper and equilateral triangles. In fact the whole site is devoted to tessellations. They even have a version of Angry Birds in a tessellation form.

I'm finishing this up for two reasons. First, there have been two power outages already this afternoon in the last 15 minutes and second, I have to finish packing for a trip I am taking to Los Angeles. I needed to get ahead so I do not miss anything.

Let me now what you think. I love hearing from people.

## Wednesday, July 26, 2017

### First Day of School.

For most people, the first day of school should be happening sometime within the next 6 weeks. My students have their first day of school on August 15th.

By the time students hit high school, teachers tend to pass out syllabus, go over the grading policy, talk about attendance, or give a test to see what students remember.

That first day of school can be a bit hard for students who may have spent the summer going to bed late and sleeping equally late in the morning. Its a sudden readjustment for both them and us.

I've read things which suggest a teacher can put off all of the usual business until the second or third day

I usually start the first day by having students work on remembering math terms. All they need is a piece of paper, a white board, a tablet, and something to write with. They need to write the alphabet vertically so A is at the top of the paper and Z is at the bottom. I give them 10 minutes to write down mathematical terms for as many letters as they can. At the end of 10 minutes, I have them place their list on the desk right in front of them.

I have everyone get up and check out everyone else's answers. They may not touch but they may look. At the end of 2 minutes, I have them sit down before giving them 2 more minutes to add to their lists.

At this point, I ask for suggestions for each letter. I always write down several choices and if someone gives a suggestion that is boarder line, I ask them for their justification before students vote to decide if its acceptable. I often run out of time before we get through the whole list.

My students enjoy it. I tell them it is a way to get their brains back into thinking about math. I never grade this activity.

One suggestion I've seen is to have students do a self portrait or write an auto biography so the teacher can learn more about them. Along the same line , have students write four clues about themselves on an index card and sign it with a code name. Gather these cards up, shuffle, and pass the cards out to the students. Challenge them to find the person whose clues they have. They become the detectives. Students will have to question each other in order to gain information.

Let the students brainstorm ideas for new years resolutions for the classroom. Resolutions should be positive and can be recorded on cloud shaped papers which are pinned to the bulletin board with the words We Resolve: What about organizing a scavenger hunt where students look for the pencil sharpener, paper, hall passes, etc so they get to know where you keep things.

Keep your eyes open for the next list of suggestions for the first day of school but it won't happen for at least a week. Let me know what you think.

By the time students hit high school, teachers tend to pass out syllabus, go over the grading policy, talk about attendance, or give a test to see what students remember.

That first day of school can be a bit hard for students who may have spent the summer going to bed late and sleeping equally late in the morning. Its a sudden readjustment for both them and us.

I've read things which suggest a teacher can put off all of the usual business until the second or third day

I usually start the first day by having students work on remembering math terms. All they need is a piece of paper, a white board, a tablet, and something to write with. They need to write the alphabet vertically so A is at the top of the paper and Z is at the bottom. I give them 10 minutes to write down mathematical terms for as many letters as they can. At the end of 10 minutes, I have them place their list on the desk right in front of them.

I have everyone get up and check out everyone else's answers. They may not touch but they may look. At the end of 2 minutes, I have them sit down before giving them 2 more minutes to add to their lists.

At this point, I ask for suggestions for each letter. I always write down several choices and if someone gives a suggestion that is boarder line, I ask them for their justification before students vote to decide if its acceptable. I often run out of time before we get through the whole list.

My students enjoy it. I tell them it is a way to get their brains back into thinking about math. I never grade this activity.

One suggestion I've seen is to have students do a self portrait or write an auto biography so the teacher can learn more about them. Along the same line , have students write four clues about themselves on an index card and sign it with a code name. Gather these cards up, shuffle, and pass the cards out to the students. Challenge them to find the person whose clues they have. They become the detectives. Students will have to question each other in order to gain information.

Let the students brainstorm ideas for new years resolutions for the classroom. Resolutions should be positive and can be recorded on cloud shaped papers which are pinned to the bulletin board with the words We Resolve: What about organizing a scavenger hunt where students look for the pencil sharpener, paper, hall passes, etc so they get to know where you keep things.

Keep your eyes open for the next list of suggestions for the first day of school but it won't happen for at least a week. Let me know what you think.

## Tuesday, July 25, 2017

### iTunes University Part 2

I had so much fun exploring this site for more math based materials one can easily integrate into the classroom.

Check out the Math of Design, a series of podcasts by Professor Jay Kapraff on 11 different topics such as the structure behind structures, and from tangrams to Amish Quilts. These podcasts are about half an hour long.

Queen Mary, University of London has the Mathematical Magic with 16 short videos on how mathematics relates to magic including two calculator tricks, dicing with destiny, and a short piece on jokers.

If you only have time for shorts, take a look at Math Snacks by New Mexico State Learning Lab. The animation shorts run anywhere from 2 to 6 minutes and address ratios in a variety of situations. The shorts are in both English and Spanish. I watched one on dating where the girl compared the number of words she spoke to the number her date spoke. It took her three dates to finally find someone who spoke the same number of words for a 1:1 ratio. The shorts are cute.

I found an hour long segment on the Mathematics of Juggling in Physical Sciences and Mathematics by Cornell University. It is one hour long and explores the mathematics involved with juggling. I have a few students who are trying to learn to juggle and they just might.

On the other hand, Open University has Mathematical Models: from Sundials to Number Engines. I watched the video on sundials which explained in detail how it works. The particular sundial used as an example showed both the time of day and time of year. I found how they used certain paths to show the time of year. It was cool. In addition, I checked out the video which discussed recording sales on clay tablets in Babylonia. Apparently, they used a base 60 system. The spokesperson indicated writing was developed for mathematical modeling. This site is more of a history of models but it was quite interesting.

Spice up a day by showing students one of three videos in Rollercoaster Design by Open University. The three videos discuss the designing of a ride called Nemesis in England from a mathematical point of view including showing a graph of the ride if you stretched it out completely. Each video has a transcript in case its needed.

For those days your students need a bit more of a challenge, check out Math Challenges by the University of Warwick which has 6 problems for students to solve. Solutions are provided so the teacher knows what to do. I looked at a problem which showed 5 glasses. The first three were filled and the last two were empty. You can only make one move so the classes are alternating, full, empty, full, empty, full. How do you do it.

These are only a few of the collections and courses offered via iTunes University. There are actual math classes such as geometry, algebra, string theory but there are also interesting ones on gliding, etc. There are even some iBooks available. Go look, check things out to decide which ones you want to spice up your class with.

Let me know what you think.

Check out the Math of Design, a series of podcasts by Professor Jay Kapraff on 11 different topics such as the structure behind structures, and from tangrams to Amish Quilts. These podcasts are about half an hour long.

Queen Mary, University of London has the Mathematical Magic with 16 short videos on how mathematics relates to magic including two calculator tricks, dicing with destiny, and a short piece on jokers.

If you only have time for shorts, take a look at Math Snacks by New Mexico State Learning Lab. The animation shorts run anywhere from 2 to 6 minutes and address ratios in a variety of situations. The shorts are in both English and Spanish. I watched one on dating where the girl compared the number of words she spoke to the number her date spoke. It took her three dates to finally find someone who spoke the same number of words for a 1:1 ratio. The shorts are cute.

I found an hour long segment on the Mathematics of Juggling in Physical Sciences and Mathematics by Cornell University. It is one hour long and explores the mathematics involved with juggling. I have a few students who are trying to learn to juggle and they just might.

On the other hand, Open University has Mathematical Models: from Sundials to Number Engines. I watched the video on sundials which explained in detail how it works. The particular sundial used as an example showed both the time of day and time of year. I found how they used certain paths to show the time of year. It was cool. In addition, I checked out the video which discussed recording sales on clay tablets in Babylonia. Apparently, they used a base 60 system. The spokesperson indicated writing was developed for mathematical modeling. This site is more of a history of models but it was quite interesting.

Spice up a day by showing students one of three videos in Rollercoaster Design by Open University. The three videos discuss the designing of a ride called Nemesis in England from a mathematical point of view including showing a graph of the ride if you stretched it out completely. Each video has a transcript in case its needed.

For those days your students need a bit more of a challenge, check out Math Challenges by the University of Warwick which has 6 problems for students to solve. Solutions are provided so the teacher knows what to do. I looked at a problem which showed 5 glasses. The first three were filled and the last two were empty. You can only make one move so the classes are alternating, full, empty, full, empty, full. How do you do it.

These are only a few of the collections and courses offered via iTunes University. There are actual math classes such as geometry, algebra, string theory but there are also interesting ones on gliding, etc. There are even some iBooks available. Go look, check things out to decide which ones you want to spice up your class with.

Let me know what you think.

## Monday, July 24, 2017

### iTunes University Part 1.

When I plan a lesson, I often forget to check iTunes University for offerings to incorporate in my math classes. There are always courses and classes being added so its worth checking out.

Over the next two days, I'm going to suggest several with interesting activities or topics for the classroom.

The first is Curious Math: Foundations of Math by Orr and Kyle Pearce. The class was designed to bridge two different math classes in Ontario. It is a combination of iBooks, 3 Act tasks, and some have a teachers manual to accompany them. This course covers measurement, proportional reasoning, graphing, linear relationships, and algebraic representations.

Some of the activities are only for iPads but most can be downloaded onto a computer so if your school has bandwidth issues, you can download many of the videos to your computer to show. I checked out a video showing a young man beating the world record for number of claps per minute. He set a new record but I cannot believe how fast he was able to clap. Over 800 claps in one minute. It was awesome. This was part of a 3 act task. I'm wanting to do it with my class.

The second is from the The University of Oxford called The Secrets of Mathematics. This series of lectures discuss a variety of topics from symmetry to modeling genes, to What maths really do for a total of 38 different lectures. Each lecture lasts about an hour, some more, some less. I began listening to one called The Sound of Symmetry which covered symmetry in nature. For me, I found it fascinating.

Other lectures like The Mathematics of Visual Illusion, or The History of Mathematics in 300 stamps, along with Maths in Music caught my eye. I want to watch these myself and learn more. In fact, I can use these lectures for learning to take visual notes before I try to explain it to my students. I have a purpose for watching these.

I will not have more than 45 minutes in a class period for watching these so if I break them in half to show, I can get through a few and expose students to a different perspective on mathematics. I think we need to do this so students see mathematics is not always solving equations per-say.

Check out the first lecture in the Beauty of Mathematics from Aspen Ideas Festival called A Mathematician Reads the newspaper. It is an hour long lecture where a mathematician shows all the ways math is used in newspapers. This might help answer "When will I ever see this?"

Open University has a wonderful class called Exploring Mathematics: A Powerful Tool which explores ways math is used in the real world. I watched a short introductory video which touched on specialized bamboo scaffolding in Hong Kong and predicting climate change. The bamboo scaffolding caught me right there. Other topics include How Math Helps Dolphins, a five minute video examining the use of statistical modeling to determine endangered species survival rates.

Open University has a second class in this nature called Exploring Mathematics: Maths in Nature and Art. Some of the topics include How a sundial works, Manufacturing patterns (designing carpets), spirals in nature, and the Lure of fractals. Every video comes with a transcript in this and the previous one. In addition, most videos are under 10 minutes long.

Tomorrow, I'm going to look at a few more classes, collections, and podcasts created to show students that math is related to real life. Let me know what you think. Have fun exploring these.

Over the next two days, I'm going to suggest several with interesting activities or topics for the classroom.

The first is Curious Math: Foundations of Math by Orr and Kyle Pearce. The class was designed to bridge two different math classes in Ontario. It is a combination of iBooks, 3 Act tasks, and some have a teachers manual to accompany them. This course covers measurement, proportional reasoning, graphing, linear relationships, and algebraic representations.

Some of the activities are only for iPads but most can be downloaded onto a computer so if your school has bandwidth issues, you can download many of the videos to your computer to show. I checked out a video showing a young man beating the world record for number of claps per minute. He set a new record but I cannot believe how fast he was able to clap. Over 800 claps in one minute. It was awesome. This was part of a 3 act task. I'm wanting to do it with my class.

The second is from the The University of Oxford called The Secrets of Mathematics. This series of lectures discuss a variety of topics from symmetry to modeling genes, to What maths really do for a total of 38 different lectures. Each lecture lasts about an hour, some more, some less. I began listening to one called The Sound of Symmetry which covered symmetry in nature. For me, I found it fascinating.

Other lectures like The Mathematics of Visual Illusion, or The History of Mathematics in 300 stamps, along with Maths in Music caught my eye. I want to watch these myself and learn more. In fact, I can use these lectures for learning to take visual notes before I try to explain it to my students. I have a purpose for watching these.

I will not have more than 45 minutes in a class period for watching these so if I break them in half to show, I can get through a few and expose students to a different perspective on mathematics. I think we need to do this so students see mathematics is not always solving equations per-say.

Check out the first lecture in the Beauty of Mathematics from Aspen Ideas Festival called A Mathematician Reads the newspaper. It is an hour long lecture where a mathematician shows all the ways math is used in newspapers. This might help answer "When will I ever see this?"

Open University has a wonderful class called Exploring Mathematics: A Powerful Tool which explores ways math is used in the real world. I watched a short introductory video which touched on specialized bamboo scaffolding in Hong Kong and predicting climate change. The bamboo scaffolding caught me right there. Other topics include How Math Helps Dolphins, a five minute video examining the use of statistical modeling to determine endangered species survival rates.

Open University has a second class in this nature called Exploring Mathematics: Maths in Nature and Art. Some of the topics include How a sundial works, Manufacturing patterns (designing carpets), spirals in nature, and the Lure of fractals. Every video comes with a transcript in this and the previous one. In addition, most videos are under 10 minutes long.

Tomorrow, I'm going to look at a few more classes, collections, and podcasts created to show students that math is related to real life. Let me know what you think. Have fun exploring these.

## Sunday, July 23, 2017

## Saturday, July 22, 2017

### Warm-up

Note: I know from personal experience parades do not move at 2.5 mph constantly. Usually, we start and stop. I wanted students to think about reasonable vs standard numerical answers.

## Friday, July 21, 2017

### Realization Concerning Fractions

The other day I pondered fractions. What are they and how do we teach this to students? Although students should know the topic well by the time they start high school, many of my students do not.

I have taken time to show students that the parts need to be of equal size within the figure but I have never explained the size is not important when showing one fractions.

By that I mean, the figure could be large or small, it does not matter as long as 1 part is shaded in. You can see I have two different sized shapes, both illustrating 1/4th.

The other way to show a fraction might be using this formation as one out of a group. The size of the individual circles should be the same but I could have used individual squares.

I do not use this representation as much because the teachers in the elementary use the first representation. I don't think they ever teach it the second way.

I know I need to include the second representation more often so they develop a better understanding.

However, when we show students addition and subtraction of fractions, we need to make sure students understand both shapes have to be the same size with equal subdivisions.

From what I've seen in the elementary classes, teachers do not take time to teach this because most of the elementary teachers I know are afraid of math and do not have an understanding themselves.

I remember years ago, one of my students had an "ah ha" moment when she realized the subdivisions within the shape had to be equal. Up to this point, she would draw the divisions any old way so they were not equal.

Imagine getting to high school, a senior at that, and having no understanding of that basic idea. This coming year I have two pre-algebra classes and one foundations of math class. I know I am going to include more illustrations in the class when teaching fractions.

I also plan to create drawings to help reinforce the idea that when adding or subtracting fractions, the size of the shape or group has to be the same so they "see" it. I've read that our brains do much better remembering material when there is a visual illustration to go with a concept.

Sorry, I'm running lazy but I had to take something into town to be cleaned, ran errands, and forgot I hadn't done this column yet. I hope everyone has fun this weekend and let me know what you think.

I have taken time to show students that the parts need to be of equal size within the figure but I have never explained the size is not important when showing one fractions.

By that I mean, the figure could be large or small, it does not matter as long as 1 part is shaded in. You can see I have two different sized shapes, both illustrating 1/4th.

The other way to show a fraction might be using this formation as one out of a group. The size of the individual circles should be the same but I could have used individual squares.

I do not use this representation as much because the teachers in the elementary use the first representation. I don't think they ever teach it the second way.

I know I need to include the second representation more often so they develop a better understanding.

However, when we show students addition and subtraction of fractions, we need to make sure students understand both shapes have to be the same size with equal subdivisions.

From what I've seen in the elementary classes, teachers do not take time to teach this because most of the elementary teachers I know are afraid of math and do not have an understanding themselves.

I remember years ago, one of my students had an "ah ha" moment when she realized the subdivisions within the shape had to be equal. Up to this point, she would draw the divisions any old way so they were not equal.

Imagine getting to high school, a senior at that, and having no understanding of that basic idea. This coming year I have two pre-algebra classes and one foundations of math class. I know I am going to include more illustrations in the class when teaching fractions.

I also plan to create drawings to help reinforce the idea that when adding or subtracting fractions, the size of the shape or group has to be the same so they "see" it. I've read that our brains do much better remembering material when there is a visual illustration to go with a concept.

Sorry, I'm running lazy but I had to take something into town to be cleaned, ran errands, and forgot I hadn't done this column yet. I hope everyone has fun this weekend and let me know what you think.

## Thursday, July 20, 2017

### Are Interactive Notebooks Effective?

Recently, I've been seeing a move towards getting rid of interactive notebooks and replacing them with digital collaboration and learning. The largest argument I've seen against it is that all students do is glue it into the notebook and that is all. They don't learn from that!

I am not sure I would fully agree with that particular idea because I get students in high school who have not been trained to take notes or to use them effectively once the notes are recorded.

I serious thought of not having them this year but realized I do not know if I'll have my iPads for another year so just in case, I will have the composition books available for notes. I found a blog whose author lists seven reasons students should use interactive notebooks.

1. They teach students to organize and synthesis their thoughts.

2. They meet multiple learning styles both in and out of the classroom.

3. When students work in their notebooks at home, parents get to see what they are doing.

4. This notebook acts as portfolio showing growth over time and is a place students can record their self reflections.

5. This interactive notebook becomes a personalized textbook which can open avenues for extended learning.

6. Students gain ownership using color and creativity.

7. They reduce clutter because its all in one place.

One new suggestion I have seen in interactive notebooks is to have students record notes on the right hand side of the page while reserving the left hand page for putting the material into their own words, pictures, etc. This suggestion fits in with my desire to teach students to create sketch notes also known as visual notes. It gives me a nice way to have them practice it starting from their notes.

It is also suggested students be given a chance to decorate the front cover of their interactive notebook so it is personalized. The first page should be left open as the table of contents so students know exactly where to find the notes on a specific topic. All left hand pages will be even numbered while the right hand pages have odd numbers. Furthermore, all pages should be numbered and dated.

In addition to putting the material into their own words or creating pictures, the left side can be used to brainstorm, show their thinking when completing a task or solving a problem. This is their side to learn that not all notes must be words.

One teacher recommended teachers create their own interactive notebook so students can consult them should they be absent or want to make sure they are up to date on notes. I've used these before but I learned a few things last year. The most important thing I learned is to have any thing that needs to be filled in located where they are easily found. I like the idea of having my own notebook so students can come in after school to fill in what they missed due to travel or an absence.

Let me know what you think. I'm working my way through a couple of books on visual note taking so I can instruct my students in this as I've read it makes it easier for them to remember material.

I am not sure I would fully agree with that particular idea because I get students in high school who have not been trained to take notes or to use them effectively once the notes are recorded.

I serious thought of not having them this year but realized I do not know if I'll have my iPads for another year so just in case, I will have the composition books available for notes. I found a blog whose author lists seven reasons students should use interactive notebooks.

1. They teach students to organize and synthesis their thoughts.

2. They meet multiple learning styles both in and out of the classroom.

3. When students work in their notebooks at home, parents get to see what they are doing.

4. This notebook acts as portfolio showing growth over time and is a place students can record their self reflections.

5. This interactive notebook becomes a personalized textbook which can open avenues for extended learning.

6. Students gain ownership using color and creativity.

7. They reduce clutter because its all in one place.

One new suggestion I have seen in interactive notebooks is to have students record notes on the right hand side of the page while reserving the left hand page for putting the material into their own words, pictures, etc. This suggestion fits in with my desire to teach students to create sketch notes also known as visual notes. It gives me a nice way to have them practice it starting from their notes.

It is also suggested students be given a chance to decorate the front cover of their interactive notebook so it is personalized. The first page should be left open as the table of contents so students know exactly where to find the notes on a specific topic. All left hand pages will be even numbered while the right hand pages have odd numbers. Furthermore, all pages should be numbered and dated.

In addition to putting the material into their own words or creating pictures, the left side can be used to brainstorm, show their thinking when completing a task or solving a problem. This is their side to learn that not all notes must be words.

One teacher recommended teachers create their own interactive notebook so students can consult them should they be absent or want to make sure they are up to date on notes. I've used these before but I learned a few things last year. The most important thing I learned is to have any thing that needs to be filled in located where they are easily found. I like the idea of having my own notebook so students can come in after school to fill in what they missed due to travel or an absence.

Let me know what you think. I'm working my way through a couple of books on visual note taking so I can instruct my students in this as I've read it makes it easier for them to remember material.

## Wednesday, July 19, 2017

### Visual Patterns

I recently heard that math is composed of patterns and the equation represents the pattern. I like that and I may put it up on my wall but why use visual patterns to teach math.

First, visual math teaches students to think about changes recursively and secondly rationally.

The other day, I came across a web site called visual patterns. The site has 240 different patterns for students to figure out the equation associated with the pattern.

Each activity shows the first three iterations of the pattern and then gives the value of the 42nd iteration so you know that total. Using the given information, people are expected to arrive at the equation producing the pattern.

This is a cool site but if you are looking for the answers, you will not find them. You will find a teachers page and a gallery with more information but you will not find all the answers.

These activities would make great warm-ups where students need to include their thinking process. While searching for ideas on Visual patterns, I stumbled across a two volume Algebra Book which uses visual patterns to teach algebra.

I looked at the first few exercises in each volume. The lessons come with everything including examples of ways students could solve the problems because students do not always use the same method to find the answers. Yes it does use blackline masters so if you are a teacher who prefers letting students explore things for themselves, it isn't hard to set up a google doc or google slides for cooperation.

The books actually go hand in hand with the visual patterns activities. I have some students who struggle with math. I think something like this may help them improve their understanding while making it a bit more interesting.

If you want students to learn more about finding the equation that expresses the pattern, Desmos has a lovely activity called "Visual Patterns Tribute" based on the first site I wrote about. It makes the whole topic a game allowing students to choose their own adventure. The activity comes with a teacher guide so the teacher can make notes as they try out the student preview. In addition, it comes with a page for students to use as they work through it but the page is more like a data sheet for students to record their findings.

There are certain topics which may be harder for us to teach because we don't understand the visual patterns. I freely admit visual patterns is not something used when I went through school. It was never even mentioned when I went through my teacher's training class so I have no idea how to teach quadratics using visual representation.

MATA has two links on this page to teach linear and quadratic functions using visual patterns. What I loved about the links is on the quadratic one, they included one or two for x^2 + 1 and even more complex quadratics. Once I saw these, my mind went "Yes" so I have a way of using this in class for specific topics.

If you'd like to know about possible student misunderstanding for these types of problems check out this blog entry as it discusses this topic.

I love finding new tools for my teaching arsenal. Let me know what you think! I love hearing from people. Have a great day.

First, visual math teaches students to think about changes recursively and secondly rationally.

The other day, I came across a web site called visual patterns. The site has 240 different patterns for students to figure out the equation associated with the pattern.

Each activity shows the first three iterations of the pattern and then gives the value of the 42nd iteration so you know that total. Using the given information, people are expected to arrive at the equation producing the pattern.

This is a cool site but if you are looking for the answers, you will not find them. You will find a teachers page and a gallery with more information but you will not find all the answers.

These activities would make great warm-ups where students need to include their thinking process. While searching for ideas on Visual patterns, I stumbled across a two volume Algebra Book which uses visual patterns to teach algebra.

I looked at the first few exercises in each volume. The lessons come with everything including examples of ways students could solve the problems because students do not always use the same method to find the answers. Yes it does use blackline masters so if you are a teacher who prefers letting students explore things for themselves, it isn't hard to set up a google doc or google slides for cooperation.

The books actually go hand in hand with the visual patterns activities. I have some students who struggle with math. I think something like this may help them improve their understanding while making it a bit more interesting.

If you want students to learn more about finding the equation that expresses the pattern, Desmos has a lovely activity called "Visual Patterns Tribute" based on the first site I wrote about. It makes the whole topic a game allowing students to choose their own adventure. The activity comes with a teacher guide so the teacher can make notes as they try out the student preview. In addition, it comes with a page for students to use as they work through it but the page is more like a data sheet for students to record their findings.

There are certain topics which may be harder for us to teach because we don't understand the visual patterns. I freely admit visual patterns is not something used when I went through school. It was never even mentioned when I went through my teacher's training class so I have no idea how to teach quadratics using visual representation.

MATA has two links on this page to teach linear and quadratic functions using visual patterns. What I loved about the links is on the quadratic one, they included one or two for x^2 + 1 and even more complex quadratics. Once I saw these, my mind went "Yes" so I have a way of using this in class for specific topics.

If you'd like to know about possible student misunderstanding for these types of problems check out this blog entry as it discusses this topic.

I love finding new tools for my teaching arsenal. Let me know what you think! I love hearing from people. Have a great day.

## Tuesday, July 18, 2017

### Renting to Own.

If you watch television or read the newspaper, you'll see ads hawking the opportunity to rent furniture or appliances for a minimum weekly rental price. If you rent the item long enough, the company states you now own it.

It sounds like a great deal. For a minimal cost you can get that refrigerator or television you want but can't afford to buy. Unfortunately, these places do not tell you how much the total cost will be so you cannot do a full comparison.

This video is a great way to introduce the topic. It shows a man who steps into a rent to own store and checks out the cost. It was fun watching the guy figure out the cost of buying vs renting. He calculated it was going to cost three times more to rent than buy.

To give students a better idea of the interest rates charged for people to buy through leasing. This site gives some excellent examples of prices if purchased at a store vs purchased through the rent to own companies. It goes into detail and explains why these companies can charge exorbitant rates when leasing to people.

There are occasions when renting is a great option such as wanting to rent a large flat screen television for the super bowl game or needing tables for a wedding. For short term rentals of a week or two, this is a valid option. If you want to own something, this is not the way to go.

What makes it attractive to people is that most of these places do not require a credit check. This means if someone has gone through bankruptcy or have lousy credit, they can rent items from these companies. For people who do not have a lot of money available, renting to own is attractive because you know how much your weekly payment but too many do not stop to check out the final cost.

This brochure discusses rent to own covering things like the store owns the merchandise until you finish paying it off. If you miss a payment, they can take it back and you lose all your money. It also gives a couple of examples showing the cost of an item if you buy it vs using rent to own to purchase it. It is a well designed and easy to read.

This blog has a wonderful activity designed to have students calculate the amount of interest charged by rent to own companies. In the first part author shows students how to calculate the interest. The second part has students select three items and calculate the interest themselves. If you follow this with this activity. The activity has walks students through an activity designed to lead student through figuring out the cost of rent to own.

I think this is an important topic to expose students to because many students in the lower socioeconomic level use rent to own all the time. Let me know what you think!

It sounds like a great deal. For a minimal cost you can get that refrigerator or television you want but can't afford to buy. Unfortunately, these places do not tell you how much the total cost will be so you cannot do a full comparison.

This video is a great way to introduce the topic. It shows a man who steps into a rent to own store and checks out the cost. It was fun watching the guy figure out the cost of buying vs renting. He calculated it was going to cost three times more to rent than buy.

To give students a better idea of the interest rates charged for people to buy through leasing. This site gives some excellent examples of prices if purchased at a store vs purchased through the rent to own companies. It goes into detail and explains why these companies can charge exorbitant rates when leasing to people.

There are occasions when renting is a great option such as wanting to rent a large flat screen television for the super bowl game or needing tables for a wedding. For short term rentals of a week or two, this is a valid option. If you want to own something, this is not the way to go.

What makes it attractive to people is that most of these places do not require a credit check. This means if someone has gone through bankruptcy or have lousy credit, they can rent items from these companies. For people who do not have a lot of money available, renting to own is attractive because you know how much your weekly payment but too many do not stop to check out the final cost.

This brochure discusses rent to own covering things like the store owns the merchandise until you finish paying it off. If you miss a payment, they can take it back and you lose all your money. It also gives a couple of examples showing the cost of an item if you buy it vs using rent to own to purchase it. It is a well designed and easy to read.

This blog has a wonderful activity designed to have students calculate the amount of interest charged by rent to own companies. In the first part author shows students how to calculate the interest. The second part has students select three items and calculate the interest themselves. If you follow this with this activity. The activity has walks students through an activity designed to lead student through figuring out the cost of rent to own.

I think this is an important topic to expose students to because many students in the lower socioeconomic level use rent to own all the time. Let me know what you think!

## Monday, July 17, 2017

### Skateboard Parks.

I am visiting my parents and family so I get to see things I don't usually see because of where I live. This past Saturday, I walked from the farmers market to the library and then over to the shopping center.

As I walked from the library, behind the police station, I really paid attention to the skateboard park that had been built in the spare land right behind the library.

I realized there are arcs and angles all through the park. The arcs provide the curved ramps you see skateboarders whiz up so they can execute a turn in the air before heading down.

As I researched the topic, it appears they requirement for ramps depends on the type of trick being performed. A quarter pipe ramp has an arc that is 1/4 of a sphere while a half pipe is two quarter ramps connected by a flat area or uses 1/2 of a sphere. There are also spines which are quarter pipes put back to back. These are often put in the middle of half pipes. A vert ramp is a quarter pipe with a higher vertical back.

It was interesting reading how the curve is actually created. After consulting a variety of do it yourself sites, I finally figured out how they do it. Take a 4 by 8 sheet of plywood. Fix a 6 foot tall board to one side. At the top, place a non-stretch string that is also 6 feet tall with a marker at the end. Then pulling the string taunt, use the marker to draw the curve on the plywood and you have your curve. In theory, you have a curve with a radius of 3 feet.

I am not a skateboarder. There is no skateboarding park out where I am but it appears to me, the circle is removed to create the curve on the ramps. I wonder if my students can picture this. So what is the best way to help students use this information?

I believe the starting point is to have students research information on each type of ramp and how the curves are created. They might also want to determine whether a 3 or 4 foot radius is better for these ramps. Once they have this information, they can design their ideal skateboard park. This site has great information on what a designer keeps in mind as they create a skateboard park so the information in here could be used to provide parameters of their design.

Before they can turn their design in, they need to explain why they designed the park the way they did. Since I do not skateboard, it might be they want more quarter pipes or more half pipes but I'd want to know their logic involved in their design.

Let me know what you think! I can't even stand on a skateboard without falling off so I have no idea but I do have a few students who own a skateboard they can ride down a walkway. I think they might find this interesting.

As I walked from the library, behind the police station, I really paid attention to the skateboard park that had been built in the spare land right behind the library.

I realized there are arcs and angles all through the park. The arcs provide the curved ramps you see skateboarders whiz up so they can execute a turn in the air before heading down.

As I researched the topic, it appears they requirement for ramps depends on the type of trick being performed. A quarter pipe ramp has an arc that is 1/4 of a sphere while a half pipe is two quarter ramps connected by a flat area or uses 1/2 of a sphere. There are also spines which are quarter pipes put back to back. These are often put in the middle of half pipes. A vert ramp is a quarter pipe with a higher vertical back.

It was interesting reading how the curve is actually created. After consulting a variety of do it yourself sites, I finally figured out how they do it. Take a 4 by 8 sheet of plywood. Fix a 6 foot tall board to one side. At the top, place a non-stretch string that is also 6 feet tall with a marker at the end. Then pulling the string taunt, use the marker to draw the curve on the plywood and you have your curve. In theory, you have a curve with a radius of 3 feet.

I am not a skateboarder. There is no skateboarding park out where I am but it appears to me, the circle is removed to create the curve on the ramps. I wonder if my students can picture this. So what is the best way to help students use this information?

I believe the starting point is to have students research information on each type of ramp and how the curves are created. They might also want to determine whether a 3 or 4 foot radius is better for these ramps. Once they have this information, they can design their ideal skateboard park. This site has great information on what a designer keeps in mind as they create a skateboard park so the information in here could be used to provide parameters of their design.

Before they can turn their design in, they need to explain why they designed the park the way they did. Since I do not skateboard, it might be they want more quarter pipes or more half pipes but I'd want to know their logic involved in their design.

Let me know what you think! I can't even stand on a skateboard without falling off so I have no idea but I do have a few students who own a skateboard they can ride down a walkway. I think they might find this interesting.

## Sunday, July 16, 2017

## Saturday, July 15, 2017

## Friday, July 14, 2017

### Real Life Discounts.

If you wonder what happened to yesterday's entry, I am visiting family and took the day off to visit with them. I'm back.

I am on a list which sends out daily e-mails suggesting eBooks I might be interested in purchasing. These are books that might be $7.99 normally but are on sale for $0.99.

This is a perfect to use in the math class when discussing discounts. The ads do not give the percent discount so it is easy to have students calculate it.

To take it one step further, students could go to various online sites such as amazon, wal-mart, or others to shop for items they'd like. Most sites list the manufacturers suggested retail price and the price the item is being sold at. Some even give a percent discount. Students can calculate the percent discount if not listed, check the site's accuracy for discount, and compare which site is a better buy for each item, or see which site is the best site to purchase everything from.

Throw in a discussion of manufacturers suggested retail price(MSRP) vs the actual price charged, the items which are generally sold using the MSRP vs those that are generally offered at less than the MSRP. Most books are sold at the cover price which is the manufacturers suggested retail price unless you belong to a plan which gives an automatic discount or things are on sale.

I love applying the MSRP to purchasing automobiles and other vehicles because the sales price is always compared to the MSRP but they never seem to give the discount as a percent, only as an amount. I think most people are excited by the fact you get money off they don't look at the actual percent.

The reason I chose online is because they list more information on the ad than the stores have on the shelf. In addition, it is fun to take the price the object online and compare it to the price the same item costs in the store. It it really cheaper to order it or is it cheaper locally. For me, in my situation, its often easier to order from someone than to try to find it in town but every situation is different.

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

I am on a list which sends out daily e-mails suggesting eBooks I might be interested in purchasing. These are books that might be $7.99 normally but are on sale for $0.99.

This is a perfect to use in the math class when discussing discounts. The ads do not give the percent discount so it is easy to have students calculate it.

To take it one step further, students could go to various online sites such as amazon, wal-mart, or others to shop for items they'd like. Most sites list the manufacturers suggested retail price and the price the item is being sold at. Some even give a percent discount. Students can calculate the percent discount if not listed, check the site's accuracy for discount, and compare which site is a better buy for each item, or see which site is the best site to purchase everything from.

Throw in a discussion of manufacturers suggested retail price(MSRP) vs the actual price charged, the items which are generally sold using the MSRP vs those that are generally offered at less than the MSRP. Most books are sold at the cover price which is the manufacturers suggested retail price unless you belong to a plan which gives an automatic discount or things are on sale.

I love applying the MSRP to purchasing automobiles and other vehicles because the sales price is always compared to the MSRP but they never seem to give the discount as a percent, only as an amount. I think most people are excited by the fact you get money off they don't look at the actual percent.

The reason I chose online is because they list more information on the ad than the stores have on the shelf. In addition, it is fun to take the price the object online and compare it to the price the same item costs in the store. It it really cheaper to order it or is it cheaper locally. For me, in my situation, its often easier to order from someone than to try to find it in town but every situation is different.

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

## Wednesday, July 12, 2017

### Splat

Steve Wybourney, author of The Writing On The Classroom Wall, shared 50 regular splats and 20 fraction splats. A splat is one or more ink blots which cover a certain number of dots. The idea is for the student to use the information provided to find an answer.

Each activity is set up as a power point but works as well in keynote. The activity walks the student through each step On the first activity of each bundle guides students through the process so they know what is expected. One step includes having students think of possible combinations or totals which is one way to have students develop number sense.

Towards the end of each splat, the answers are revealed but only after students have a chance to think about their answers including the opportunity to explain their thinking.

The fraction splats operate in the same manner except students see a combination of whole and factional circles. They look at the total before a splat covers some of them. It is up to the student to workout the fraction covered by the splat.

I will be teaching a basic math course for students who are well below grade level and a pre-algebra for students who are only a little below grade level. I can see using these for both classes as a way for students to develop number sense. Most of mine arrive in high school without having developed number sense.

I can hardly wait to use them. In the meantime check them out and let me know what you think.

## Tuesday, July 11, 2017

### Why Use Consistant Units.

As I mentioned in yesterday's column, Puerto Rico does not use consistent units. This definitely

leads to confusion if you are not used to this. Many of my students see no reason for making sure

they all use the same units of measurement.

One reason for using the same units is to increase communication so everyone is on the same page. We have all heard stories of something two companies built but the final product did not work because the two companies used different systems.

The perfect example can be found if you look at the Mars Climate Orbiter from 1998. The Jet Propulsion Laboratory (JPL) in Pasadena worked with Lockheed Martin Aeronautics in Denver on the software. Unfortunately, JPL did all their calculations in metric while Lockheed Martin used the English system of feet, inches, etc.

This lead to the space craft entered the Mars atmosphere at the wrong angle and was lost. What makes this so sad, is the fact neither group caught the error in all the months they worked together. This error resulted in a 125 million dollar ship being destroyed.

If you look back to 1983, you will find the story of a Jet plane running out of fuel due to a mistake in converting units. Air Canada's Boeing 767 ran out of fuel a routine flight because someone thought the fuel weight was in kilograms when it was actually in pounds. As you know 1 kilogram = 2.2 lbs. This means the plane had half the fuel it should have had. Fortunately, the two pilots onboard had gliding experience and were able to safely land the plane.

This is not a recent occurrence. It has been happening since the Vasa Warship was built back in the 1600's. Apparently the ship was asymmetrical because builders used the 12 inch Swedish ruler while the other side used the Austrian 11 inch ruler and these folks were working in the same shipyard!

The final example is the Laufenberg bridge built back in the early 2000's. The bridge straddles Germany and Switzerland. The problem arose because each country defines sea level. This caused one side to to be 54 cm higher than the other side. In other words one side was taller than the other.

So the next time students wonder about the importance of using the same units, you can share these examples. Let me know what you think.

## Monday, July 10, 2017

### Crazy Math

As you know, I've been traveling and currently I am in San Juan, Puerto Rico for a conference. The conference is almost over and I'll be hitting the road again, heading for Washington state.

In the math classes, I stress using all one system of measurement to make sure all units are consistent but I got a shock the other day on the way out to the radio telescope in Aerocibo, Puerto Rico.

As the bus traveled on the toll road, I noticed all the markers along the side of the road were in kilometers. We use them to check mileage along any road so if we have to call in for assistance, we can tell them we are at mile 35.7 or at kilometer 22.3. Signs showing the distance to the next town are also in kilometers but the speed limit signs are in miles per hour!

You read that right. Speed limits are in miles per hour which can make things really confusing. I had to ask to make sure it was miles per hour and not kilometer per hour because the posted speed limit was 65 and 55. No unit designation on it.

The guy sitting next to me on the bus was as confused as me when we saw those numbers. Although, Puerto Rico is a territory of the United States, it mixes English measurement with metric. For instance, gas is sold per liter while tomatoes are per pound. My taxi driver tried to tell me gas was only $1.29 per gallon yet sold for 60 cents per liter. I don't think he knows the real conversion rate. He also said each gallon has 5 liters.

I checked out some real estate adds for houses in the area. The main unit is meters squared with some translation into standard measurements but I'm not sure if the unit is for the house or the land. That part wasn't that clear. I did not watch any local television stations so I do not know if temperatures are given in Celsius or Fahrenheit.

When school starts in the fall, I can share the unique mix of systems with my students. I plan to take time to explain how confusing it was to see things in one or the other or both depending. This is the first place I've ever been where I've seen both systems used.

Let me know what you think. Puerto Rico appears to be the exception to the rule. Have a good day and enjoy yourself.

In the math classes, I stress using all one system of measurement to make sure all units are consistent but I got a shock the other day on the way out to the radio telescope in Aerocibo, Puerto Rico.

As the bus traveled on the toll road, I noticed all the markers along the side of the road were in kilometers. We use them to check mileage along any road so if we have to call in for assistance, we can tell them we are at mile 35.7 or at kilometer 22.3. Signs showing the distance to the next town are also in kilometers but the speed limit signs are in miles per hour!

You read that right. Speed limits are in miles per hour which can make things really confusing. I had to ask to make sure it was miles per hour and not kilometer per hour because the posted speed limit was 65 and 55. No unit designation on it.

The guy sitting next to me on the bus was as confused as me when we saw those numbers. Although, Puerto Rico is a territory of the United States, it mixes English measurement with metric. For instance, gas is sold per liter while tomatoes are per pound. My taxi driver tried to tell me gas was only $1.29 per gallon yet sold for 60 cents per liter. I don't think he knows the real conversion rate. He also said each gallon has 5 liters.

I checked out some real estate adds for houses in the area. The main unit is meters squared with some translation into standard measurements but I'm not sure if the unit is for the house or the land. That part wasn't that clear. I did not watch any local television stations so I do not know if temperatures are given in Celsius or Fahrenheit.

When school starts in the fall, I can share the unique mix of systems with my students. I plan to take time to explain how confusing it was to see things in one or the other or both depending. This is the first place I've ever been where I've seen both systems used.

Let me know what you think. Puerto Rico appears to be the exception to the rule. Have a good day and enjoy yourself.

## Sunday, July 9, 2017

## Saturday, July 8, 2017

## Friday, July 7, 2017

### Different Breakeven Points

I am in Puerto Rico, attending a conference. As part of pre-conference events, I took a tour of the radio telescope at Arecibo.

About 50 of us boarded a charted bus and traveled about 1.5 hours to the site. Along the way, I listened in to a conversation in which two people discussed breakeven points of conferences and other events.

As they discussed this topic, my brain sort of went Dah! because any event has to breakeven before making a profit.

This is something my students can relate to because when they become juniors and seniors, their class takes over running the concession stand at school. They have to stock the concession stand with sodas, chips, and other snacks. To do this, they usually get a copy of the Span Alaska catalogue, prepare an order, send it off before waiting several weeks for the supplies to arrive.

Although they use the money in their class account to purchase the supplies, I don't think their advisor has ever taken time to explain how to calculate profit. Just think, the breakeven point for this is when they've earned the money to cover their original expenditure. A part of the original expenditure includes the shipping to get it all out to the village. To make this more relevant, I plan to have students calculate the amount of profit they will make by determining the amount of money they could make if they sell everything at certain prices.

If I take this idea a step further, I could have students figure out the cost of everything for the prom, including food, entertainment, and supplies. Then they have to decide how many people need to attend to breakeven based on the price of tickets. Then they need to look at projections of possible profit based on the number of people attending based on price.

Students at my school tend to charge the same as in the others in the past. I do not believe students look at the price of tickets to project attendance so they know what they should charge for maximum profit. Most times, class advisors do not have students do any type of projections. The standard way is to order the supplies and sell without any calculations or thought.

In other areas, students could use different types of events such as dances to do the same types of calculation. If we tie breakeven to events students are more familiar with, maybe they will understand the concept better since these are situations they relate to or have prior knowledge.

Monday, I'll be off to another topic. I hope you all have a wonderful day. Take care and let me know what you think.

About 50 of us boarded a charted bus and traveled about 1.5 hours to the site. Along the way, I listened in to a conversation in which two people discussed breakeven points of conferences and other events.

As they discussed this topic, my brain sort of went Dah! because any event has to breakeven before making a profit.

This is something my students can relate to because when they become juniors and seniors, their class takes over running the concession stand at school. They have to stock the concession stand with sodas, chips, and other snacks. To do this, they usually get a copy of the Span Alaska catalogue, prepare an order, send it off before waiting several weeks for the supplies to arrive.

Although they use the money in their class account to purchase the supplies, I don't think their advisor has ever taken time to explain how to calculate profit. Just think, the breakeven point for this is when they've earned the money to cover their original expenditure. A part of the original expenditure includes the shipping to get it all out to the village. To make this more relevant, I plan to have students calculate the amount of profit they will make by determining the amount of money they could make if they sell everything at certain prices.

If I take this idea a step further, I could have students figure out the cost of everything for the prom, including food, entertainment, and supplies. Then they have to decide how many people need to attend to breakeven based on the price of tickets. Then they need to look at projections of possible profit based on the number of people attending based on price.

Students at my school tend to charge the same as in the others in the past. I do not believe students look at the price of tickets to project attendance so they know what they should charge for maximum profit. Most times, class advisors do not have students do any type of projections. The standard way is to order the supplies and sell without any calculations or thought.

In other areas, students could use different types of events such as dances to do the same types of calculation. If we tie breakeven to events students are more familiar with, maybe they will understand the concept better since these are situations they relate to or have prior knowledge.

Monday, I'll be off to another topic. I hope you all have a wonderful day. Take care and let me know what you think.

## Thursday, July 6, 2017

### Break Even Points

I got to wondering about cost per units, breakeven points, and profit. This takes things one step further than just figuring out cost per unit of objects when shopping.

The reality is: we do teach these as linear equations in general but we provide the formulas and we ask the questions. We seldom let them explore equations to find the information themselves.

I do not believe most of my students have a good understanding of breakeven points or of the equation itself. They can identify the parts but I think they can do the math without really understanding the topic.

I plan to create a short unit where I have students research the cost of purchasing a game app. There are several websites where a person can purchase the script for a game with the understanding they will make a few changes so they are not simply reselling the original game. This is part of the startup cost. Next they have to figure out how much they want to sell the new app for so they can find the rest of equation so they can calculate the number of units they need to sell to find breakeven point.

Once they have the breakeven point, I will throw in the idea they do not get all of the money every time an app is sold. I want to introduce they pay a percentage to apple store or google store for them to handle all purchases so the return they actually get is less than what they charge so that might change the cost per unit and change the breakeven point.

This is important because my students have no knowledge of how a person decides the unit cost especially when looking at a product such as a game app for mobile devices. My students think they get to keep every bit of money the app is sold for. In addition the start up cost means very little to students out in the Bush of Alaska because most people who start a business do so in an old abandoned building or use part of a room. They don't see any of that as part of the start up cost.

Once the students have the new equation and breakeven point, I want to throw in the idea of having to add more games because most games have a length of interest before people want to move on to another game. I honestly believe this will make the idea of cost, revenue, and breakeven more relevant and understandable. Let me know what you think.

The reality is: we do teach these as linear equations in general but we provide the formulas and we ask the questions. We seldom let them explore equations to find the information themselves.

I do not believe most of my students have a good understanding of breakeven points or of the equation itself. They can identify the parts but I think they can do the math without really understanding the topic.

I plan to create a short unit where I have students research the cost of purchasing a game app. There are several websites where a person can purchase the script for a game with the understanding they will make a few changes so they are not simply reselling the original game. This is part of the startup cost. Next they have to figure out how much they want to sell the new app for so they can find the rest of equation so they can calculate the number of units they need to sell to find breakeven point.

Once they have the breakeven point, I will throw in the idea they do not get all of the money every time an app is sold. I want to introduce they pay a percentage to apple store or google store for them to handle all purchases so the return they actually get is less than what they charge so that might change the cost per unit and change the breakeven point.

This is important because my students have no knowledge of how a person decides the unit cost especially when looking at a product such as a game app for mobile devices. My students think they get to keep every bit of money the app is sold for. In addition the start up cost means very little to students out in the Bush of Alaska because most people who start a business do so in an old abandoned building or use part of a room. They don't see any of that as part of the start up cost.

Once the students have the new equation and breakeven point, I want to throw in the idea of having to add more games because most games have a length of interest before people want to move on to another game. I honestly believe this will make the idea of cost, revenue, and breakeven more relevant and understandable. Let me know what you think.

## Wednesday, July 5, 2017

### Mercury and Apollo Space Crafts.

I just attended a conference in Phoenix which was strong on science. One of the things I attended, included information on various space craft and a bit on the actual astronauts.

As you know the first astronauts were actually rather small in they could not be over 5 foot 11 and could weigh no more than 180 lbs due to the size of the Mercury Capsule.

The capsule was only 6 ft 10 in tall and 6 ft 2.5 inches in diameter. I think it would be great to have students calculate the amount of volume in a cylinder of that size. I realize the capsule is not a cylinder but using a cylinder makes it easier. It gives an idea of the space an astronaut lived in for several days. It would not be that hard to have students create a cylinder out of construction paper or cardboard to provide a visual representation to give students a better idea of size.

Apollo 11 was a lot bigger at 10 ft 7 in by 12 ft 10 inches but it had three people instead of only one in the Mercury. Again, it is easy to have students calculate the volume but they could take it a step further to determine if the amount of space per person is the same or smaller? If it is less per person, they can calculate the percent difference with space per person.

If you want to take it further, look at the International Space Section which is 356 feet by 240 feet. What is the volume of this craft which houses up to 10 people at any one time. How many square feet are allowed per person and how does it compare to the mercury capsule?

I realize that the volume calculated for each space craft includes electronics, seats and other objects but I want to have students calculate volume, and space per person before having them brain storm on why just calculating volume per person can be slightly misleading. I want them to do research to see if they can find the information on the amount of space planned for the astronauts.

I am off to load. I am on my way to Puerto Rico and will be arriving there a bit later today. Have a good day.

As you know the first astronauts were actually rather small in they could not be over 5 foot 11 and could weigh no more than 180 lbs due to the size of the Mercury Capsule.

The capsule was only 6 ft 10 in tall and 6 ft 2.5 inches in diameter. I think it would be great to have students calculate the amount of volume in a cylinder of that size. I realize the capsule is not a cylinder but using a cylinder makes it easier. It gives an idea of the space an astronaut lived in for several days. It would not be that hard to have students create a cylinder out of construction paper or cardboard to provide a visual representation to give students a better idea of size.

Apollo 11 was a lot bigger at 10 ft 7 in by 12 ft 10 inches but it had three people instead of only one in the Mercury. Again, it is easy to have students calculate the volume but they could take it a step further to determine if the amount of space per person is the same or smaller? If it is less per person, they can calculate the percent difference with space per person.

If you want to take it further, look at the International Space Section which is 356 feet by 240 feet. What is the volume of this craft which houses up to 10 people at any one time. How many square feet are allowed per person and how does it compare to the mercury capsule?

I realize that the volume calculated for each space craft includes electronics, seats and other objects but I want to have students calculate volume, and space per person before having them brain storm on why just calculating volume per person can be slightly misleading. I want them to do research to see if they can find the information on the amount of space planned for the astronauts.

I am off to load. I am on my way to Puerto Rico and will be arriving there a bit later today. Have a good day.

## Tuesday, July 4, 2017

## Monday, July 3, 2017

### Cost of Travel.

I am currently in Phoenix as I write today's entry. When I purchased my tickets, I had to decide which routing I wanted. If you do enough travel, you know you usually have so many different choices on routing and cost.

What factors do people look at when deciding on selecting the ticket. Some routes, you really don't have much choice because there are only two or three flights a day, while other routes have tons of choices.

In addition, you often have to look at arrival times. I travel to a place where the last flight arrives at midnight but the car rental places shut down by 9 PM. So I end up renting a hotel room for the night, pick up the car the next morning and I'm off.

I usually look for the cheapest routing but tonight I wondered if the cheapest ticket is actually the cheapest? Can we look at just the total cost or can we look at the cost based on the price per mile? Should one include the cost of the hotel room as part of the cost?

I travel with an airline who offers more perks with each level gained. When you get to 20,000 miles, you earn the status of MVP or Most Valued Passenger. I usually get this level due to living in Alaska. Every time I fly to Seattle, I get between 1500 and 2000 miles depending on the routing. At 40,000 miles, you become MVP Gold and if you manage 75,000 miles you become a MVP Platinum. I've never gotten that high but I have managed MVP Gold a couple of years. Gold gives you some First Class upgrades you can use on purchased tickets but you are also bumped up to First Class if there is space.

So you may wonder where I am going with this? If the cheapest really the cheapest or if you paid a bit more and got 500 miles more is that better. Being the geek I am, I've been known to sit down and calculate the cost per mile to see which one was actually a better price. I also look at the additional cost of the miles if I pay for a longer routing.

Later this summer, I am traveling to Finland but I have to fly from Alaska to DC before I head off to Finland. I have to do this because there are two flights out of Anchorage each week but the days do not work for getting back to work on August 14th so I had to look at alternatives. The cost was not much more going that routing and I get a bunch more miles on my frequent flyer account.

I want my students plan a trip, choose their flights, and provide their reasons for selecting said routing. I also want them to calculate the cost per mile to see if there is much of a difference between choices.

Let me know what you think. Have a great day and a wonderful 4th of July celebration. Be back on Wednesday with more math.

What factors do people look at when deciding on selecting the ticket. Some routes, you really don't have much choice because there are only two or three flights a day, while other routes have tons of choices.

In addition, you often have to look at arrival times. I travel to a place where the last flight arrives at midnight but the car rental places shut down by 9 PM. So I end up renting a hotel room for the night, pick up the car the next morning and I'm off.

I usually look for the cheapest routing but tonight I wondered if the cheapest ticket is actually the cheapest? Can we look at just the total cost or can we look at the cost based on the price per mile? Should one include the cost of the hotel room as part of the cost?

I travel with an airline who offers more perks with each level gained. When you get to 20,000 miles, you earn the status of MVP or Most Valued Passenger. I usually get this level due to living in Alaska. Every time I fly to Seattle, I get between 1500 and 2000 miles depending on the routing. At 40,000 miles, you become MVP Gold and if you manage 75,000 miles you become a MVP Platinum. I've never gotten that high but I have managed MVP Gold a couple of years. Gold gives you some First Class upgrades you can use on purchased tickets but you are also bumped up to First Class if there is space.

So you may wonder where I am going with this? If the cheapest really the cheapest or if you paid a bit more and got 500 miles more is that better. Being the geek I am, I've been known to sit down and calculate the cost per mile to see which one was actually a better price. I also look at the additional cost of the miles if I pay for a longer routing.

Later this summer, I am traveling to Finland but I have to fly from Alaska to DC before I head off to Finland. I have to do this because there are two flights out of Anchorage each week but the days do not work for getting back to work on August 14th so I had to look at alternatives. The cost was not much more going that routing and I get a bunch more miles on my frequent flyer account.

I want my students plan a trip, choose their flights, and provide their reasons for selecting said routing. I also want them to calculate the cost per mile to see if there is much of a difference between choices.

Let me know what you think. Have a great day and a wonderful 4th of July celebration. Be back on Wednesday with more math.

## Sunday, July 2, 2017

## Saturday, July 1, 2017

## Friday, June 30, 2017

### Trajectories

Do you realize how often students use an instinctual knowledge of math to calculate the trajectory path for the game they are playing? Watch them! They use it so much.

I've done it in when playing Angry Birds. I aim and hope but depending on whether its too high or low, I readjust by eyeballing it. After a while I become quite good at it but not perfect.

My students do the same thing with their games involving some sort of projectile. They try again and again until they get it. If the score is too low, they redo it just like me.

My first thought is they are using some sort of quadratic but programmers may not think that way when they create an app. So if they don't use the quadratic, what do they use? Think physics with the two vectors that help locate the position of the object along its trajectory. These games use the projectile motion formulas and trajectory formulas.

The various formulas are as follows:

Horizontal distance = the velocity along the x-axis * time.

Horizontal velocity = the initial velocity along the x - axis.

Vertical distance = the velocity along the y-axis * time minus 1/2 gt^2

Vertical velocity = the initial velocity along the y-axis minus gt.

Then there are these more specific ones:

Time of flight is t = (2*Vo*Sinx)/g where Vo is the initial velocity and sinx is component along the y axis and is usually in degrees. G is gravity or 9.8 m/s/s

Maximum Height is H = (Vo^2* Sin^2x)/2g

Range is R = (Vo^2*Sin2x)/g.

This site has a great example of the coding to show how programmers code projectile motion in games such as Angry Birds. This type of activity, especially if you provide the coding and the math, students can compare and see how the two relate. A programmer I know is always telling me that one should "borrow" already written code rather than starting from scratch as it is faster.

I would love to incorporate this type of project in my trig class so they experience a real life situation that uses trig. In addition, it would be good to show substitutions and such but unfortunately, we are not allowed to teach coding in school because it is not seen as an important skill. I disagree but can't do anything about it.

Let me know your thoughts on the topic. I love hearing from people.

I've done it in when playing Angry Birds. I aim and hope but depending on whether its too high or low, I readjust by eyeballing it. After a while I become quite good at it but not perfect.

My students do the same thing with their games involving some sort of projectile. They try again and again until they get it. If the score is too low, they redo it just like me.

My first thought is they are using some sort of quadratic but programmers may not think that way when they create an app. So if they don't use the quadratic, what do they use? Think physics with the two vectors that help locate the position of the object along its trajectory. These games use the projectile motion formulas and trajectory formulas.

The various formulas are as follows:

Horizontal distance = the velocity along the x-axis * time.

Horizontal velocity = the initial velocity along the x - axis.

Vertical distance = the velocity along the y-axis * time minus 1/2 gt^2

Vertical velocity = the initial velocity along the y-axis minus gt.

Then there are these more specific ones:

Time of flight is t = (2*Vo*Sinx)/g where Vo is the initial velocity and sinx is component along the y axis and is usually in degrees. G is gravity or 9.8 m/s/s

Maximum Height is H = (Vo^2* Sin^2x)/2g

Range is R = (Vo^2*Sin2x)/g.

This site has a great example of the coding to show how programmers code projectile motion in games such as Angry Birds. This type of activity, especially if you provide the coding and the math, students can compare and see how the two relate. A programmer I know is always telling me that one should "borrow" already written code rather than starting from scratch as it is faster.

I would love to incorporate this type of project in my trig class so they experience a real life situation that uses trig. In addition, it would be good to show substitutions and such but unfortunately, we are not allowed to teach coding in school because it is not seen as an important skill. I disagree but can't do anything about it.

Let me know your thoughts on the topic. I love hearing from people.

## Thursday, June 29, 2017

### Student Interaction.

As teachers we are having to look at changing the type of lessons we use in class. If you watch students outside of class, they are usually sitting down, against a wall, playing some sort of games. It might be a mideaval protect a town type game, one of the angry bird manifestations, or something like Zombies vs Plants.

A game in which they are active. They are not listening to a lecture. They are not filling out a worksheet. They are not passive.

They are thinking about strategy. They are active. They are working towards a goal. Unfortunately, most of us do not teach in a way that capitalizes on their abilities. I started google classroom the final quarter of school but I hadn't figured out how to make things more interactive than using digitized worksheets.

Unfortunately, most of the interactive activities I found on the internet are all basically the same. It involves printed material a student reads in order to fill out a worksheet. More of the same passive style work but students need something more. A few activities require the use of manipulatives but it still requires filling out a worksheet.

This summer, I'm taking several different classes. One is designed to create interactive lessons with videos, activities, and fewer read in order to fill out the worksheet activities. The others are so I can use Google Suite for Education and create a classroom that is more student centered than teacher centered.

In addition, I'm reading books and getting ideas such as creating trailers to tease students with the next topic just like movie trailers tease the population into wanting to see the movies. Another book is helping me think about incorporating gaming into the classroom but not just using badges. Setting the unit up as a Mission Impossible experience with a mission so they feel as if they are in charge.

So many ideas from so many different books. I am realistic enough to know, I cannot implement everything I've learned but I can choose a few ideas and implement them at the beginning of school. Add a couple more ideas each quarter, through the year. The next year, add a few more. I hope the ideas I implement will help students take ownership of their learning so I do less and let them do more. Will it work? I don't know.

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

A game in which they are active. They are not listening to a lecture. They are not filling out a worksheet. They are not passive.

They are thinking about strategy. They are active. They are working towards a goal. Unfortunately, most of us do not teach in a way that capitalizes on their abilities. I started google classroom the final quarter of school but I hadn't figured out how to make things more interactive than using digitized worksheets.

Unfortunately, most of the interactive activities I found on the internet are all basically the same. It involves printed material a student reads in order to fill out a worksheet. More of the same passive style work but students need something more. A few activities require the use of manipulatives but it still requires filling out a worksheet.

This summer, I'm taking several different classes. One is designed to create interactive lessons with videos, activities, and fewer read in order to fill out the worksheet activities. The others are so I can use Google Suite for Education and create a classroom that is more student centered than teacher centered.

In addition, I'm reading books and getting ideas such as creating trailers to tease students with the next topic just like movie trailers tease the population into wanting to see the movies. Another book is helping me think about incorporating gaming into the classroom but not just using badges. Setting the unit up as a Mission Impossible experience with a mission so they feel as if they are in charge.

So many ideas from so many different books. I am realistic enough to know, I cannot implement everything I've learned but I can choose a few ideas and implement them at the beginning of school. Add a couple more ideas each quarter, through the year. The next year, add a few more. I hope the ideas I implement will help students take ownership of their learning so I do less and let them do more. Will it work? I don't know.

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

## Wednesday, June 28, 2017

### EDPuzzle

I recently got a chance to play with EDPuzzle, a web based site which allows a teacher to select a video, shorten it, add questions, assessments, etc so the student has to respond while watching the video.

I logged on the other day using a google account and since I'd not used it before, it provided a tutorial. In just a few minutes I had my first interactive video completed. I thought I would share it with everyone.

After watching the final product, I noticed I could have put a couple questions at bit later but on the other hand, the first question is a good one to explore their prior knowledge. Questions can require an answer, open ended or multiple choice.

I love the idea of assigning an interactive video to the students. When students watch the video, it will automatically stop at the question. It does not give them a choice to skip. Students must submit an answer before it allows them to continue the video. In addition, they do have the option to rewatch the part of the video, the question applies to.

This is a screen shot of the question. It asks about the material just presented in the video. As you see the only two choices are to submit an answer or rewatch. If they push the submit button in the hopes of moving on, it won't let them. It provides the message "Please, answer all the questions." It will not let them go on without putting even IDK ("I don't know") in.

This means the students will do more than just "watch" a video. They have to really pay attention to the material and they can work at their own pace rather than at someone else's including mine. The questions can be used to signal the information is important so perhaps they might want to make it a part of their notes.

It allows you to assign the activity to your students via google classroom. In addition, its easy to see who has watched the video, their grade, when they last worked on the activity and when it was turned in. This information can be exported as a Numbers document.

You can also look at the individual questions to see who answered each question and how it was answered so you know who needs additional scaffolding, who should redo the assignment. The information can also be exported in Numbers so you can see all the results.

I am impressed with this website. It does not take long to find a video, cut it to the exact length, add the questions and assign it. Just a few minutes. Something I can accomplish in my prep period. In fact, I can get a few done during my prep period.

I made the video using my iPad so I could not use the audio track or audio notes due to my lack of flash but I can use them on my computer. There is an app for students to use to access any activities. As long as they have a Google ID, they can sign in. Not a problem.

I've been taking a class this summer where I learn to create interactive lessons and this is a perfect addition to the lessons because they are expected to be active in their own learning and this is a great way to do it.

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

I logged on the other day using a google account and since I'd not used it before, it provided a tutorial. In just a few minutes I had my first interactive video completed. I thought I would share it with everyone.

After watching the final product, I noticed I could have put a couple questions at bit later but on the other hand, the first question is a good one to explore their prior knowledge. Questions can require an answer, open ended or multiple choice.

I love the idea of assigning an interactive video to the students. When students watch the video, it will automatically stop at the question. It does not give them a choice to skip. Students must submit an answer before it allows them to continue the video. In addition, they do have the option to rewatch the part of the video, the question applies to.

This is a screen shot of the question. It asks about the material just presented in the video. As you see the only two choices are to submit an answer or rewatch. If they push the submit button in the hopes of moving on, it won't let them. It provides the message "Please, answer all the questions." It will not let them go on without putting even IDK ("I don't know") in.

This means the students will do more than just "watch" a video. They have to really pay attention to the material and they can work at their own pace rather than at someone else's including mine. The questions can be used to signal the information is important so perhaps they might want to make it a part of their notes.

It allows you to assign the activity to your students via google classroom. In addition, its easy to see who has watched the video, their grade, when they last worked on the activity and when it was turned in. This information can be exported as a Numbers document.

You can also look at the individual questions to see who answered each question and how it was answered so you know who needs additional scaffolding, who should redo the assignment. The information can also be exported in Numbers so you can see all the results.

I am impressed with this website. It does not take long to find a video, cut it to the exact length, add the questions and assign it. Just a few minutes. Something I can accomplish in my prep period. In fact, I can get a few done during my prep period.

I made the video using my iPad so I could not use the audio track or audio notes due to my lack of flash but I can use them on my computer. There is an app for students to use to access any activities. As long as they have a Google ID, they can sign in. Not a problem.

I've been taking a class this summer where I learn to create interactive lessons and this is a perfect addition to the lessons because they are expected to be active in their own learning and this is a great way to do it.

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

## Tuesday, June 27, 2017

### Visual Note Taking

I've been seeing quite a few things on visual note taking. I'm impressed with the drawings and the scribbled words that make visual notes. I've seen some wonderful examples but not for math.

Of course, I'm always interested in how to use something like this in my classroom. The idea is to attach both images and text together to utilize dual coding.

The first step is to look at visual note taking itself. Visual note taking is a way to take notes using pictures, doodles, shapes, lines, and text on paper. This type of note taking can personalize learning, synthesis information, and helps create meaning for the students when they use their own ideas.

The value is in making the notes, not in being a perfect artist because the notes are designed by the individual. Encourage students to use sketches, pictures, doodles, shapes, line, and text to create their notes. It is important to practice before having students begin creating notes for class.

Start students off with simple topics so they can practice taking visual notes. As they practice, move them towards taking math notes so they have the chance to develop their own style.

According to this blog entry, visual note taking is good for lessons with categories and subcategories or topics with layers, or relationships between ideas and topics. There are three different possibilities for doing this.

The first is for putting together the pieces showing relationships with parts and connections. One can use pieces/puzzles or gears, webs, Vann Diagrams, or clusters. They are great for showing how parts connect.

The second is for showing steps or processes. Ladders, steps, pyramids and stacks are great for showing this type of topic. The shapes help students remember better.

Third is telling stories to connect material using metaphors, call outs or posters.

I think students have to learn to create their own visual notes for math, including being able to look at a textbook and change the written words into pictures which make sense to them. I plan to revisit this topic later on. I'm leaving Friday night for Phoenix and then San Juan, Puerto Rico before visiting my parents. I'll keep posting but you might get some information as I stumble across it at my conferences.

Let me know what you think. Have a good day.

Of course, I'm always interested in how to use something like this in my classroom. The idea is to attach both images and text together to utilize dual coding.

The first step is to look at visual note taking itself. Visual note taking is a way to take notes using pictures, doodles, shapes, lines, and text on paper. This type of note taking can personalize learning, synthesis information, and helps create meaning for the students when they use their own ideas.

The value is in making the notes, not in being a perfect artist because the notes are designed by the individual. Encourage students to use sketches, pictures, doodles, shapes, line, and text to create their notes. It is important to practice before having students begin creating notes for class.

Start students off with simple topics so they can practice taking visual notes. As they practice, move them towards taking math notes so they have the chance to develop their own style.

According to this blog entry, visual note taking is good for lessons with categories and subcategories or topics with layers, or relationships between ideas and topics. There are three different possibilities for doing this.

The first is for putting together the pieces showing relationships with parts and connections. One can use pieces/puzzles or gears, webs, Vann Diagrams, or clusters. They are great for showing how parts connect.

The second is for showing steps or processes. Ladders, steps, pyramids and stacks are great for showing this type of topic. The shapes help students remember better.

Third is telling stories to connect material using metaphors, call outs or posters.

I think students have to learn to create their own visual notes for math, including being able to look at a textbook and change the written words into pictures which make sense to them. I plan to revisit this topic later on. I'm leaving Friday night for Phoenix and then San Juan, Puerto Rico before visiting my parents. I'll keep posting but you might get some information as I stumble across it at my conferences.

Let me know what you think. Have a good day.

## Monday, June 26, 2017

### Self Reflection.

As a teacher, it is strongly recommended we stop and think about our lessons. How they went? What could I have done differently? How can I make it better? That is one reason I always read books and try to find ways to engage my students more. I do not take time to have my students reflect on their understanding of the current material.

What would it do if we had students reflect on their learning? Would it help us clarify and reteach material sooner than if we waited for a quiz or test? Unfortunately, if I do not provide guidance to help them learn to write their thoughts down, I'll get many who will just copy things from their notes.

This site has a wonderful question with ideas on how to answer the question. The question she used "How did you improve as a Mathematician today?" is so much better than "What did you learn?" or "What do you remember?". Both questions would get me "nothing" as the answer. She offers five suggestions for students to write on.

1. Describe a new strategy you learned today.

2. Tell a math word you learned today and what it means.

3. Describe a mistake you made and what you learned from it.

4. Explain how you challenged yourself today.

5. Tell about something you noticed today and how it helped you solve a problem.

These are great questions to help guide students in answering the overall question. The self reflection can actually help students change their mindset from fixed to one of growth.

Other ways to encourage self reflection are:

1. Have students keep math journals based on certain math prompts. The above five suggestions would work quite well with the journal. In addition, it is quite easy to find an assortment of math prompts. I figure it is quite easy to do via technology by setting up a google doc for each student and then checking each doc to grade.

2. Implement Student Led Conferences in which students bring their assessments and reflect on. The idea is to have the student determine where their weaknesses and strengths lay so they can focus on the topics they need to work on. This is actually a nice way to work on figuring out where you need to differentiate so as to meet the student needs.

3. Anchor charts. After a formative assessment or test, have a classroom discussion to get student feedback on problems they had trouble with or questions on. The concerns form the basis of the discussion on the process to solve each problem and the mathematical process itself.

4. Blogging. Use blogging to have students explain how they solved a problem, their thoughts, and problems they had completing the problem. When they blog, they have to slow down and think about the words they need.

These are some great ways to encourage students to reflect on their learning. I will tell you, I don't do it because I have not had any idea of how to do it. Let me know if you have any suggestions on this topic.

What would it do if we had students reflect on their learning? Would it help us clarify and reteach material sooner than if we waited for a quiz or test? Unfortunately, if I do not provide guidance to help them learn to write their thoughts down, I'll get many who will just copy things from their notes.

This site has a wonderful question with ideas on how to answer the question. The question she used "How did you improve as a Mathematician today?" is so much better than "What did you learn?" or "What do you remember?". Both questions would get me "nothing" as the answer. She offers five suggestions for students to write on.

1. Describe a new strategy you learned today.

2. Tell a math word you learned today and what it means.

3. Describe a mistake you made and what you learned from it.

4. Explain how you challenged yourself today.

5. Tell about something you noticed today and how it helped you solve a problem.

These are great questions to help guide students in answering the overall question. The self reflection can actually help students change their mindset from fixed to one of growth.

Other ways to encourage self reflection are:

1. Have students keep math journals based on certain math prompts. The above five suggestions would work quite well with the journal. In addition, it is quite easy to find an assortment of math prompts. I figure it is quite easy to do via technology by setting up a google doc for each student and then checking each doc to grade.

2. Implement Student Led Conferences in which students bring their assessments and reflect on. The idea is to have the student determine where their weaknesses and strengths lay so they can focus on the topics they need to work on. This is actually a nice way to work on figuring out where you need to differentiate so as to meet the student needs.

3. Anchor charts. After a formative assessment or test, have a classroom discussion to get student feedback on problems they had trouble with or questions on. The concerns form the basis of the discussion on the process to solve each problem and the mathematical process itself.

4. Blogging. Use blogging to have students explain how they solved a problem, their thoughts, and problems they had completing the problem. When they blog, they have to slow down and think about the words they need.

These are some great ways to encourage students to reflect on their learning. I will tell you, I don't do it because I have not had any idea of how to do it. Let me know if you have any suggestions on this topic.

## Sunday, June 25, 2017

## Saturday, June 24, 2017

## Friday, June 23, 2017

### Forensics.

As you know, I find the math in forensics quite interesting. It is a way of finding information from the evidence to find the perpetrator.

On all those shows, they find the guilty person within one hour, well 45 minutes give or take a couple minutes.

In reality it can take up to a year or longer. There was a murder in the village I live in. A young lady was stabbed multiple times and left behind the clinic. The police came out, interviewed everyone, gathered data, found evidence, and took forever but almost a year later, they arrested a young man for the crime. I shared one way I'd used math to prove a person not guilty of reckless driving.

Lets look at some other ways math is used in forensics to help solve crimes.

1. One can tell an animal hair from a human hair by calculating the ratio of the diameter of medulla to the diameter of the whole hair. If it is .5 or higher, it is animal hair. If it is lower than .5, it is human hair.

2. Estimating the size of an individual grain of pollen using the magnification of a microscope. The idea is to estimate how many grains of pollen will fit across and divide the size of the field of view by that and voila, you have the individual size.

3. Blood splatter uses trigonometric calculations using the height and distance to find the angle of impact. height is opposite while distance to the splatter is adjacent.

4. Using angles, they can tell if the pelvic bones are from a male or female. If the angle beneath the ischia bones is less than 90 degrees (forms an acute angle), it is male. If it is greater than 90 degrees ( forms an obtuse angle), its a female.

5. Time of death, looking at the drop in the body temperature. For the first 12 hours, the body cools by 1.4 degrees F each hour. After 12 hours, heat loss is calculated at .7 degrees F each hour.

6. Time of death based on insect development. They can use the development of certain insects to approximate a time of death because insects require a certain temperature to hatch and progress from one stage to another.

If the math is packaged in something as exciting as helping to solve a crime, students are more willing to do the calculations because it is a fun and applicable situation. I'll give some sites for creating a unit next week.

Let me know what you think.

On all those shows, they find the guilty person within one hour, well 45 minutes give or take a couple minutes.

In reality it can take up to a year or longer. There was a murder in the village I live in. A young lady was stabbed multiple times and left behind the clinic. The police came out, interviewed everyone, gathered data, found evidence, and took forever but almost a year later, they arrested a young man for the crime. I shared one way I'd used math to prove a person not guilty of reckless driving.

Lets look at some other ways math is used in forensics to help solve crimes.

1. One can tell an animal hair from a human hair by calculating the ratio of the diameter of medulla to the diameter of the whole hair. If it is .5 or higher, it is animal hair. If it is lower than .5, it is human hair.

2. Estimating the size of an individual grain of pollen using the magnification of a microscope. The idea is to estimate how many grains of pollen will fit across and divide the size of the field of view by that and voila, you have the individual size.

3. Blood splatter uses trigonometric calculations using the height and distance to find the angle of impact. height is opposite while distance to the splatter is adjacent.

4. Using angles, they can tell if the pelvic bones are from a male or female. If the angle beneath the ischia bones is less than 90 degrees (forms an acute angle), it is male. If it is greater than 90 degrees ( forms an obtuse angle), its a female.

5. Time of death, looking at the drop in the body temperature. For the first 12 hours, the body cools by 1.4 degrees F each hour. After 12 hours, heat loss is calculated at .7 degrees F each hour.

6. Time of death based on insect development. They can use the development of certain insects to approximate a time of death because insects require a certain temperature to hatch and progress from one stage to another.

If the math is packaged in something as exciting as helping to solve a crime, students are more willing to do the calculations because it is a fun and applicable situation. I'll give some sites for creating a unit next week.

Let me know what you think.

## Thursday, June 22, 2017

### Board Foot

I am currently working with a family member to finish off the basement. He and his wife asked to move into the basement so they could save money and help finish the basement.

All I can say is the basement has a cement floor, lots of studs and pipes for things. It does have a few power boxes but I need more in there.

While looking over some of those magazines on finishing this or that, I stumbled across a comment on board feet. I'd forgotten about that measurement. My father, being a a shop teacher, spoke about board foot and linear foot.

Technically, a board foot is a volume of 144 cubed units, such as 12 in by 12 in by 1 in while a linear foot is just the length regardless of width or depth. Honestly, I'm never sure which unit hardware stores sell the lumber by. The last ones I bought were sold for a flat rate, the board already cut to a certain length. In addition, if the boards thickness is less than 3/4 inch, it is sold by the linear foot.

After some research, I'm told some stores do sell lumber by the board foot while others sell lumber by the linear foot. Using this information and other pieces, its possible to come up with some questions students can explore.

1. If the volume of a board foot is 144 inches square, how many different measurements can a board foot have? Which ones are more likely than other? Explain your choices.

2. If you have a board that is 8 inches wide, 8 feet long, and 1 inch thick, how many board feet is that?

3. If you have a board that is 2 inches wide, 3 inches deep, 8 feet long and costs $5.00 per board foot, how much will the board cost?

4. If the board you have is 8 inches wide, 3/4 inch thick and 8 feet long, how many board feet is it? What would it cost if the hardware store charges $4.75 per board inch.

So a real life application of board feet. let me know what you think? Have a good day.

All I can say is the basement has a cement floor, lots of studs and pipes for things. It does have a few power boxes but I need more in there.

While looking over some of those magazines on finishing this or that, I stumbled across a comment on board feet. I'd forgotten about that measurement. My father, being a a shop teacher, spoke about board foot and linear foot.

Technically, a board foot is a volume of 144 cubed units, such as 12 in by 12 in by 1 in while a linear foot is just the length regardless of width or depth. Honestly, I'm never sure which unit hardware stores sell the lumber by. The last ones I bought were sold for a flat rate, the board already cut to a certain length. In addition, if the boards thickness is less than 3/4 inch, it is sold by the linear foot.

After some research, I'm told some stores do sell lumber by the board foot while others sell lumber by the linear foot. Using this information and other pieces, its possible to come up with some questions students can explore.

1. If the volume of a board foot is 144 inches square, how many different measurements can a board foot have? Which ones are more likely than other? Explain your choices.

2. If you have a board that is 8 inches wide, 8 feet long, and 1 inch thick, how many board feet is that?

3. If you have a board that is 2 inches wide, 3 inches deep, 8 feet long and costs $5.00 per board foot, how much will the board cost?

4. If the board you have is 8 inches wide, 3/4 inch thick and 8 feet long, how many board feet is it? What would it cost if the hardware store charges $4.75 per board inch.

So a real life application of board feet. let me know what you think? Have a good day.

## Wednesday, June 21, 2017

### Calculating Speed from Skid Marks.

I have always loved watching television shows which involve some sort of forensics. CSI and all its variations including NCIS captured my attention because of solving a mystery based only on evidence.

One topic they don't usually discuss is determining the speed of a vehicle based on the length of the skid marks left behind.

I have a friend who was driving home from church one day. He came around the curve, just across the train tracks, when he hit a street sweeper that was making a U-turn in the middle of the road. What's worse, there was a sign posted before the curve advising people to look for the flagman who was absent.

He was issued a ticket for speeding. He came back as soon as he got a tape measure to determine the length of the marks. He brought the information to me so I could check the officer's conclusion. So after a bit of research and calculations, I discovered he was only going about 36.5 mph in a 35 zone.

I used the calculation S = sqrt(30*D*f*n). S means speed, 30 is a constant, D is the distance of the drag marks, f refers to the drag factor based on type of road, and n is the breaking efficiency in a percent.

The officer was notified of his calculation error. I thought that would be the end of it but the officer came back with a charge of reckless driving, a charge that could result in my friend spending time in jail and loosing his license. Add injury to insult, no lawyer would touch the case because they said the case was too absurd to be prosecuted.

The city refused to provide photos of the damage to the street cleaner. They would not talk to the insurance company, or do much at all. He supplied everything he could from my calculations to photos of the damage to his car, to drawings and anything else he could think of. The insurance company was using my calculations, his pictures but the city refused to discuss it at all.

He finally went in to talk to a District Attorney to discuss his plea of not guilty with the damaged bumper in his hand. Fortunately, the D.A. had a enough classes in physics to understand my friends argument on why he was not recklessly driving. The DA basically threw the charges out due.

This is my real life example of how my use of math in real life saved a friend from getting convicted of a fairly serious charge to being freed. Mathematical equations do work within a real life context. I am going to have fun having students do this in class in the fall.

Let me know what you think. Have a good day.

One topic they don't usually discuss is determining the speed of a vehicle based on the length of the skid marks left behind.

I have a friend who was driving home from church one day. He came around the curve, just across the train tracks, when he hit a street sweeper that was making a U-turn in the middle of the road. What's worse, there was a sign posted before the curve advising people to look for the flagman who was absent.

He was issued a ticket for speeding. He came back as soon as he got a tape measure to determine the length of the marks. He brought the information to me so I could check the officer's conclusion. So after a bit of research and calculations, I discovered he was only going about 36.5 mph in a 35 zone.

I used the calculation S = sqrt(30*D*f*n). S means speed, 30 is a constant, D is the distance of the drag marks, f refers to the drag factor based on type of road, and n is the breaking efficiency in a percent.

The officer was notified of his calculation error. I thought that would be the end of it but the officer came back with a charge of reckless driving, a charge that could result in my friend spending time in jail and loosing his license. Add injury to insult, no lawyer would touch the case because they said the case was too absurd to be prosecuted.

The city refused to provide photos of the damage to the street cleaner. They would not talk to the insurance company, or do much at all. He supplied everything he could from my calculations to photos of the damage to his car, to drawings and anything else he could think of. The insurance company was using my calculations, his pictures but the city refused to discuss it at all.

He finally went in to talk to a District Attorney to discuss his plea of not guilty with the damaged bumper in his hand. Fortunately, the D.A. had a enough classes in physics to understand my friends argument on why he was not recklessly driving. The DA basically threw the charges out due.

This is my real life example of how my use of math in real life saved a friend from getting convicted of a fairly serious charge to being freed. Mathematical equations do work within a real life context. I am going to have fun having students do this in class in the fall.

Let me know what you think. Have a good day.

## Tuesday, June 20, 2017

### Mapping Patterns in Crime.

Mathematicians love the beauty of patterns. There are patterns in traffic flow, in nature, in crime and math is used to map those patterns showing where the crimes have been committed.

Crime analysis looks at the patterns of crimes being committed in which areas to determine the best response.

According to an article published by UCLA, criminals are hunter gathers who hunt for criminal activities. They follow certain patterns.

Using mathematical modeling, the police are able to find hot spots, determine the type of hot spot, and determine the best reaction to the activity. There are two types of hot spots, the first is characterized by a small rise in activity that grows while the second is a large spike in a central location. By knowing which type of hot spot, the police can respond appropriately so they do not cause the hot spot to move to another area without suppressing it.

In addition, they can determine if the it is a hot spot of violent crimes, burglary, or auto. Mathematical modeling that provides this detailed information. Apparently when a hot spot occurs for a specific crime, the chances of it occurring increase because the criminals appear to be comfortable working that area.

There are others who are investigating this type of analysis. In the Boston area, three people including a person at MIT created a program to look at trends in crimes and discovered trends that had previously been unidentified using traditional methods. In addition, they found several crimes that met the criteria but had not been classified prior to this.

Crime mapping provides information for three type of analysis. The first, tactical analysis, looks at the short term such as a crime spree because they want to stop what is going on. It looks at one criminal with many targets or one target with many criminals. It is used when an immediate response is needed.

The second, strategic crime analysis, looks at both long term and on going events. It often focuses on areas with high crime rates and tries to find ways to decrease the crime rates. The final, administrative crime analysis, looks at the police and their deployment to determine if they are being used effectively.

The data used for these crime analysis usually come from 911 call records. The crime is entered into the data base, if the perpetrator is arrested, if he is convicted, or put in jail, all of this is put into the data base. The information is analyzed using mathematical modeling so one of the three type of analysis can be applied to the data.

Let me know what you think. Have a good day and enjoy yourselves.

Crime analysis looks at the patterns of crimes being committed in which areas to determine the best response.

According to an article published by UCLA, criminals are hunter gathers who hunt for criminal activities. They follow certain patterns.

Using mathematical modeling, the police are able to find hot spots, determine the type of hot spot, and determine the best reaction to the activity. There are two types of hot spots, the first is characterized by a small rise in activity that grows while the second is a large spike in a central location. By knowing which type of hot spot, the police can respond appropriately so they do not cause the hot spot to move to another area without suppressing it.

In addition, they can determine if the it is a hot spot of violent crimes, burglary, or auto. Mathematical modeling that provides this detailed information. Apparently when a hot spot occurs for a specific crime, the chances of it occurring increase because the criminals appear to be comfortable working that area.

There are others who are investigating this type of analysis. In the Boston area, three people including a person at MIT created a program to look at trends in crimes and discovered trends that had previously been unidentified using traditional methods. In addition, they found several crimes that met the criteria but had not been classified prior to this.

Crime mapping provides information for three type of analysis. The first, tactical analysis, looks at the short term such as a crime spree because they want to stop what is going on. It looks at one criminal with many targets or one target with many criminals. It is used when an immediate response is needed.

The second, strategic crime analysis, looks at both long term and on going events. It often focuses on areas with high crime rates and tries to find ways to decrease the crime rates. The final, administrative crime analysis, looks at the police and their deployment to determine if they are being used effectively.

The data used for these crime analysis usually come from 911 call records. The crime is entered into the data base, if the perpetrator is arrested, if he is convicted, or put in jail, all of this is put into the data base. The information is analyzed using mathematical modeling so one of the three type of analysis can be applied to the data.

Let me know what you think. Have a good day and enjoy yourselves.

## Monday, June 19, 2017

### Cartography and Math

Cartographers or map makers use quite a bit of math in the creation of maps. I suspect if you asked most students "What math is used in map making?" they'd respond with a shrug, or its only used in the key in the corner.

In truth, cartographers use quite a bit of math. They use math in map scales, coordinate systems, and map projection to begin with. The math scales shows the relationship between distance on a map and distance in real life as a fraction or ratio.

The coordinate systems refers to the numerical representation of locations of places on the planet while map projection is a mathematical transformation of points from a curved surface to a flat surface. Did you know there are at least 18 different map projections including the Mercator which is the one most people are familiar with. They type of map projection chosen depends on what needs to be shown. This site has a great explanation of all the different types of map projections.

Back to the coordinate systems used in cartography. One is the geographic coordinate system which is based on longitude and latitude to pinpoint the exact location of any place on earth. The other type is a projected coordinate plane which takes the earths curved surface and projects it onto a coordinate system.

New Zealand Maths has a nice unit created which has students creating their own maps of the classroom complete with scale and coordinate planes to mark the location of an object on the map. The nice thing about this activity relates magnetic north to true north.

I can hear my students telling me that paper maps are out of fashion. Maps are on their phones, so that information is not up to date but contrary to that opinion, they are wrong. Math is even more important because mathematical equations referred to as mathematical exact visualization are what allow you to move your view of the map around, check out streets as if you are driving down them or keep track of the various labels of building, hotels, etc.

These mathematical equations are needed so the viewer can move digital maps around and still return back to your location. Tomorrow, I'm going to look into the use of maps and math to find patterns in crime.

Let me know what you think.

In truth, cartographers use quite a bit of math. They use math in map scales, coordinate systems, and map projection to begin with. The math scales shows the relationship between distance on a map and distance in real life as a fraction or ratio.

The coordinate systems refers to the numerical representation of locations of places on the planet while map projection is a mathematical transformation of points from a curved surface to a flat surface. Did you know there are at least 18 different map projections including the Mercator which is the one most people are familiar with. They type of map projection chosen depends on what needs to be shown. This site has a great explanation of all the different types of map projections.

Back to the coordinate systems used in cartography. One is the geographic coordinate system which is based on longitude and latitude to pinpoint the exact location of any place on earth. The other type is a projected coordinate plane which takes the earths curved surface and projects it onto a coordinate system.

New Zealand Maths has a nice unit created which has students creating their own maps of the classroom complete with scale and coordinate planes to mark the location of an object on the map. The nice thing about this activity relates magnetic north to true north.

I can hear my students telling me that paper maps are out of fashion. Maps are on their phones, so that information is not up to date but contrary to that opinion, they are wrong. Math is even more important because mathematical equations referred to as mathematical exact visualization are what allow you to move your view of the map around, check out streets as if you are driving down them or keep track of the various labels of building, hotels, etc.

These mathematical equations are needed so the viewer can move digital maps around and still return back to your location. Tomorrow, I'm going to look into the use of maps and math to find patterns in crime.

Let me know what you think.

## Sunday, June 18, 2017

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