## Sunday, April 30, 2017

## Saturday, April 29, 2017

## Friday, April 28, 2017

### SpreadSheets in Math.

Like it or not, spread sheets are an important part of running a business. Spread sheets are used to create so many documents but the bottom line is many applications within the spread sheet requires math. In addition, spreadsheets can be used for assessments in the math classroom.

Education world states spreadsheets help facilitate inquiry and analysis, explore elementary algebra as they use functions and function tables.

In addition, students can observe patterns as values are changed, can do data analysis with dynamic graphs which adjust as data is added or changed, or they can compare and contrast real life data with theoretical data. Research indicates that the use of spreadsheets in the classroom is a step towards creating student centered classroom.

What is so cool, is the use of spread sheets for sequences such as arithmetic, geometric, triangular, squared, cubed, and perhaps even the Fibonacci sequence. Create formulas for conversions of length, weight, money conversions. etc. Alphabetize lists or sort by numerical value, make dynamic magic squares, create function machines, etc.

Spreadsheets can be used to perform standard operations of multiplication, division, addition, and subtraction along with exponents, order of operations, and functions such as sum, average, minimum, and maximum. Spreadsheets are also great for finding the line of best fit when analyzing data.

According to one article I read, spreadsheets are awesome for "What If" scenarios when you want to see what certain changes do to the whole situation. For instance, if you purchase a house using an adjustable rate mortgage, what does a 1/8th percent increase do to the overall cost of the loan? How much will it change the monthly payment?. What happens if the interest drops? How does an increase or decrease change the over all cost of a 15 vs 30 year mortgage.

One suggestion from Alice Keeler who is a google guru and recently co-wrote a book on this topic is you create a vocabulary list of words for the topic in one column. Label several other columns with sub topics so students can sort the words into the proper group.

If you are not sure where to start, check out What If Math. They have a whole list of labs students can utilize in the classroom beginning with introductory labs designed to get them started. Dr Christie has another great site with links to interactive spreadsheet activities for more ideas. While you're at it check Kutztown Univeristy for more great ideas including weighted averages, creating charts, keeping a checkbook and other activities.

Remember, if you do not know how to do something, check YouTube to help you learn, then post the link in Google classroom or in a QR code so students can watch it themselves to learn. Remember, let them do most of it themselves.

I found directions so students could create their own quadratic factoring calculator using the spreadsheet. This is one of my projects to learn this year so I can have students do it next year. So Let me know what you think. Have a great day.

Education world states spreadsheets help facilitate inquiry and analysis, explore elementary algebra as they use functions and function tables.

In addition, students can observe patterns as values are changed, can do data analysis with dynamic graphs which adjust as data is added or changed, or they can compare and contrast real life data with theoretical data. Research indicates that the use of spreadsheets in the classroom is a step towards creating student centered classroom.

What is so cool, is the use of spread sheets for sequences such as arithmetic, geometric, triangular, squared, cubed, and perhaps even the Fibonacci sequence. Create formulas for conversions of length, weight, money conversions. etc. Alphabetize lists or sort by numerical value, make dynamic magic squares, create function machines, etc.

Spreadsheets can be used to perform standard operations of multiplication, division, addition, and subtraction along with exponents, order of operations, and functions such as sum, average, minimum, and maximum. Spreadsheets are also great for finding the line of best fit when analyzing data.

According to one article I read, spreadsheets are awesome for "What If" scenarios when you want to see what certain changes do to the whole situation. For instance, if you purchase a house using an adjustable rate mortgage, what does a 1/8th percent increase do to the overall cost of the loan? How much will it change the monthly payment?. What happens if the interest drops? How does an increase or decrease change the over all cost of a 15 vs 30 year mortgage.

One suggestion from Alice Keeler who is a google guru and recently co-wrote a book on this topic is you create a vocabulary list of words for the topic in one column. Label several other columns with sub topics so students can sort the words into the proper group.

If you are not sure where to start, check out What If Math. They have a whole list of labs students can utilize in the classroom beginning with introductory labs designed to get them started. Dr Christie has another great site with links to interactive spreadsheet activities for more ideas. While you're at it check Kutztown Univeristy for more great ideas including weighted averages, creating charts, keeping a checkbook and other activities.

Remember, if you do not know how to do something, check YouTube to help you learn, then post the link in Google classroom or in a QR code so students can watch it themselves to learn. Remember, let them do most of it themselves.

I found directions so students could create their own quadratic factoring calculator using the spreadsheet. This is one of my projects to learn this year so I can have students do it next year. So Let me know what you think. Have a great day.

## Thursday, April 27, 2017

### Previewing Material With Trailers.

I'm working my way through the book

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

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

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

Here is the video:

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

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

Let me know what you think.

**Teaching Like A Pirate.**I really enjoy the book because the ideas are conveyed through questions. Questions phrased in such a way as to allow your mind to come up with solutions.One question in there lead me to consider using movie trailers, like those you see on television which try to capture your interest.

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

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

Here is the video:

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

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

Let me know what you think.

## Wednesday, April 26, 2017

### Slide Rule Followup

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Tuesday, April 25, 2017

### Animation

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

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

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

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

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

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

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

## Monday, April 24, 2017

### Old Fashioned Technology

By Joe Haupt from USA |

I've been working my way through the book

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

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

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

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

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

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

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

## Sunday, April 23, 2017

## Saturday, April 22, 2017

## Friday, April 21, 2017

## Thursday, April 20, 2017

### Taking Movie Notes

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

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

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

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

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

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

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

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

Check back tomorrow, for suggestions on using videos effectively.

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

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

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

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

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

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

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

Check back tomorrow, for suggestions on using videos effectively.

## Wednesday, April 19, 2017

### Reading and Creating Math Problems.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Tuesday, April 18, 2017

### Spacing Practice

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

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

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

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

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

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

4. Keep returning to content considered important.

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

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

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

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

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

6.

## Monday, April 17, 2017

### Memory Vs Thinking

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

The quote "

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

Why else would we teach students to follow processes when solving problems rather than looking for connections.

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

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

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

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

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

The quote "

**We rely on memory rather than thinking"**from Hattie and Yates*, Visible Learning and the science of how we learn.*This quote perfectly identifies the overwhelming thought of teaching as it has been for so many years and seems to be continuing even now.

Why else would we teach students to follow processes when solving problems rather than looking for connections.

*I wonder if this is why so many of our students are unable to transfer knowledge from one situation to another or why they barely remember yesterday's lesson.*When teaching students to solve word problems, we often tell them to look for key words which may nor may not provide the correct information, make a plan, etc. Most of my student just look at the numbers, hoping to choose the correct operation. They don't stop to think about what the problems is asking or how they might solve it.

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

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

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

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

## Sunday, April 16, 2017

## Saturday, April 15, 2017

## Friday, April 14, 2017

### Relating topics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Thursday, April 13, 2017

### Followup on Reading Standards.

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

I began by reviewing these terms:

1. Main character

2. Setting

3. Event or what was happening

4. Inferences which need to be made

5. Summary - a sentence or two to summarize all the information and the answer.

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

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

We decided the main characters were the 6 bluebirds and one baby bear.

The setting: its happening by the bird feeder.

The event: splitting the seeds up evenly.

Inferences: 6 birds plus one bear means 7 totally. Splitting means division.

I had the students do the math so we had an answer for the problem

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

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

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

I began by reviewing these terms:

1. Main character

2. Setting

3. Event or what was happening

4. Inferences which need to be made

5. Summary - a sentence or two to summarize all the information and the answer.

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

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

We decided the main characters were the 6 bluebirds and one baby bear.

The setting: its happening by the bird feeder.

The event: splitting the seeds up evenly.

Inferences: 6 birds plus one bear means 7 totally. Splitting means division.

I had the students do the math so we had an answer for the problem

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

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

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

## Wednesday, April 12, 2017

### The Big Idea

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

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

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

1. It is a statement.

2. It is an idea that is central to learning mathematics.

3. It links everything together so it makes sense.

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

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

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

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

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

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

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

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

1. It is a statement.

2. It is an idea that is central to learning mathematics.

3. It links everything together so it makes sense.

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

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

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

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

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

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

## Tuesday, April 11, 2017

### Process vs Understanding.

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

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

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

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

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

According to a page on Understanding Mathematics at the University of Utah, you understand math if you:

a.) are able to explain mathematical concepts using simpler concepts and facts.

b.) make logical connections between different facts and concepts.

c.) see a connection between something new and something in math you know.

d.) can identify the principal of any math you do.

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

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

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

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

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

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

According to a page on Understanding Mathematics at the University of Utah, you understand math if you:

a.) are able to explain mathematical concepts using simpler concepts and facts.

b.) make logical connections between different facts and concepts.

c.) see a connection between something new and something in math you know.

d.) can identify the principal of any math you do.

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

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

## Monday, April 10, 2017

### Explaining Thinking

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Sunday, April 9, 2017

## Saturday, April 8, 2017

## Friday, April 7, 2017

### Relating Reading to Math

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Thursday, April 6, 2017

### R.A.C.E.

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

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

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

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

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

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

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

1. Students must answer the question.

2. Students need to explain their answer.

3. Students need to prove their answer is correct.

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

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

Have a good day.

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

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

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

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

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

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

1. Students must answer the question.

2. Students need to explain their answer.

3. Students need to prove their answer is correct.

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

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

Have a good day.

## Wednesday, April 5, 2017

### The Math of Mixing Car Paint.

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

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

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

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

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

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

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

Let me know what you think.

## Tuesday, April 4, 2017

### Building Metacognition in Math.

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

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

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

There are three ways to look at metacognition in math.

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

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

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

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

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

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

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

2.

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

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

There are three ways to look at metacognition in math.

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

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

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

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

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

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

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

2.

## Monday, April 3, 2017

### Pens and Math

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Sunday, April 2, 2017

## Saturday, April 1, 2017

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