This past weekend, I attended something called Costume College, held in Woodland Hills, California. For those not familiar with the area, its basically the Los Angeles Area.

That is why I didn't get my usual weekend warm-ups posted. I got absolutely busy and did not get a chance to post. The picture is of me, in my attire for the ball. Everyone shows their work by wearing costumes.

I costume in my spare time. Anyway, during one of the conversations, math came up. The lady I spoke with explained the pattern for her current outfit was only available in a smaller size. The store had sold out of the larger sizes, so she had to add 3/8th of an inch all around to make it fit.

I've had to do it but the other way. I've had to subtract 1/2 inch to make everything fit properly. Many costumers write down tons of different measurements such as the length from the back of your neck to your waist, distance across the shoulders, etc because most patterns are designed for specific measurements. I have one shoulder that is slightly lower than the other due to playing French Horn for over half my life. I know a female who has larger upper arms due to heaving batteries.

In addition, many pieces of clothing are designed using some basic shapes. For instance, those circular poodle skirts from the 1950's are based on a circle with a place for the waist. Unfortunately, most fabric is not wide enough to cut out a single circle. I use circular skirts when I belly dance so I've had a lot of experience. I am lazy so I use a string and piece of chalk to draw a fourth of a circle on a folded piece of cloth to get half the skirt. I do it twice, cut out the wait area, sew, put in zipper, and hem.

Another type of skirt can be made out of two rectangles the width of the hips for the front and back. This straight skirt uses elastic to gather the extra material around the waist while the rest of the skirt falls straight down. I often leave slits in each side to make walking easier.

Furthermore, I use two rectangles as a starting point for making harem pants or loose hang about pants. I use two wide rectangles, one for each leg, cut out the crotch part, sew it, put in some elastic and I am set.

Ethnic clothing is often put together using rectangles, squares and other shapes to create the dresses, shirts, skirts etc because cloth was woven on narrow looms. Japanese kimonos originally used material that was no more than 18 inches wide so the clothing created had to take this into account and the most efficient way of doing this was through the use of shapes.

Tomorrow, I'm going to look at the use of shapes and clothing from a different point of view. I hope you liked today's material. Let me know what you think.

## Monday, July 31, 2017

## Saturday, July 29, 2017

### Busy weekend

I am at an educational conference. The internet signal is up and down. I found out about something new I will share with everyone on Monday. I do not think I will be tomorrow so if I am not, it is because I just ran out of time.

## Friday, July 28, 2017

### Model Railroads.

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.

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.

## 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

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