Up until yesterday, I didn't know there was such a thing as math museums. I knew of art museums, car museums, train museums, historical museums, etc but never a math museum.

One I found yesterday was the 3DXM Virtual Museum filled with all sorts of mathematical art. It has material on minimal surfaces, famous surfaces, conformal maps, plane curves, fractals, space curves, polyhedral and math art.

Each topic has a variety of drawings for each type along with the mathematics involved. The pictures tend to be in color and appear three dimensional. It isn't until you get to the math art section that things really come together to show math in a different way.

I checked out the work of several artists and their work is absolutely breathtaking. The rose shaped parametric surface, the Kleinian double spiral, iso construction, and Tori reflections are spectacular and if shown out of context would be taken as straight artwork.

This site is the type of place students can go and just explore while having fun checking out the interactive elements.

Other places that are not so much museums in the traditional sense of the word but have interesting things to offer both students and teachers.

1. The Exploratorium in San Francisco offers activities that can be done with your students. One suggestion is creating a math walk locally. The site offers instructions for creating one locally for people who are not sure how to go about setting one up.

2. The Smithsonian has some nice mathematical opportunities available on line. There is a virtual exhibit on Slates, Slide rules, and Software a history of mathematical teaching in America. In addition, there are several videos on the math of prehistoric climate change, or the math involved in the fish population. There are quite a few stories and articles all associated with math. So many items to integrate into class.

3. Finally is the National Museum of Mathematics in New York City does not have much in the way of online materials but it does have a page full of videos made by several different mathematicians. If you want to see what topics are covered by checking out the map of the museum.

Have fun letting students explore some new places filled with mathematics. Let me know what you think. I'd love to hear.

## Friday, November 17, 2017

## Thursday, November 16, 2017

### Exploring Math

The idea for this column came from a tweet by Tina Cardone. The idea is to create an ongoing project filled with four elements designed to have students do one activity from each column each quarter. At the end of the quarter, the students report back.

She basically created a choice board with choices to play by doing math art, math comics, or a puzzle, research either a career, a mathematician, or find an article on math to read and summarize. There is an explain section where the student explains a new math topic discovered in the last 100 years, or explains about an ancient math system, or an unsolved math problem. The final section has the student try a math challenge problem, or problems from two other sources.

The play column is in there to show students that math can be fun and is not always answering problem after problem after problem. So if you want to create your own, where would you go to find comics, art, or puzzles that students might enjoy. You could borrow her version or you could personalize it to make it more relevant to your students.

Here are places that offer math comics that are actually rather funny.

1. Comic Math filled with samples and links to various mathematically based comics both popular and lesser known ones.

2. This list has comics based on topic.

3. This is a list of 7 places with great math comics.

Now for the art possibilities:

1. Math art for kids has some great ideas including creating a city scape out of the numbers of pi. There are 21 different suggestions listed here. Even though its for kids, I think some of my high school students would enjoy them so I think I'll try to find time to incorporate them into my class.

2. Smith Curriculum has some great art projects including the Pythagorean Snail based on the Pythagorean Theorem. I should try this in my geometry class since some of the students are more artist than anything else and would rather draw than do school work.

3. To see works check out the Virtual Math Museum with some fantastic art based on mathematics.

This site is filled with mathematical puzzles that students might find interesting.

Check this site out as it lists both male and female mathematicians and their ethnicity. Its quite a list.

My students are not ready for something like this but I like the idea of playing with mathematical art to add another layer to my classroom. Let me know what you think. I'd love to hear.

She basically created a choice board with choices to play by doing math art, math comics, or a puzzle, research either a career, a mathematician, or find an article on math to read and summarize. There is an explain section where the student explains a new math topic discovered in the last 100 years, or explains about an ancient math system, or an unsolved math problem. The final section has the student try a math challenge problem, or problems from two other sources.

The play column is in there to show students that math can be fun and is not always answering problem after problem after problem. So if you want to create your own, where would you go to find comics, art, or puzzles that students might enjoy. You could borrow her version or you could personalize it to make it more relevant to your students.

Here are places that offer math comics that are actually rather funny.

1. Comic Math filled with samples and links to various mathematically based comics both popular and lesser known ones.

2. This list has comics based on topic.

3. This is a list of 7 places with great math comics.

Now for the art possibilities:

1. Math art for kids has some great ideas including creating a city scape out of the numbers of pi. There are 21 different suggestions listed here. Even though its for kids, I think some of my high school students would enjoy them so I think I'll try to find time to incorporate them into my class.

2. Smith Curriculum has some great art projects including the Pythagorean Snail based on the Pythagorean Theorem. I should try this in my geometry class since some of the students are more artist than anything else and would rather draw than do school work.

3. To see works check out the Virtual Math Museum with some fantastic art based on mathematics.

This site is filled with mathematical puzzles that students might find interesting.

Check this site out as it lists both male and female mathematicians and their ethnicity. Its quite a list.

My students are not ready for something like this but I like the idea of playing with mathematical art to add another layer to my classroom. Let me know what you think. I'd love to hear.

## Wednesday, November 15, 2017

### Finding Errors

As I mentioned yesterday, its hard for students to find the error they made if they do not get the correct answer. I've been wondering about techniques I can include to help students learn to find their errors.

It seems that once the student has completed a problem, their mind shuts the door on it and moves on because they are finished with it and don't need to check it.

One suggestion I ran across is to have a poster in the classroom for the top 11 errors made in math calculations hung somewhere in the room so they can check it before they move on.

1. Did not distribute the outside term to both terms inside the parenthesis. This includes not distributing the negative sign with the number.

2. Multiplying by 2 instead of squaring. In other words they multiply by the exponent, instead of applying the power.

3. Adding instead of subtracting or vice versa.

4. Adding instead of multiplying or vice versa.

5. Misplacing or loosing a decimal.

6. Making a rounding error.

7. Forgetting to carry a number or to borrow.

8. Forgetting to change the inequality sign when dividing or multiplying by a negative.

9. Making a mistake when cross multiplying ratios.

10. Making a mistake when adding/subtracting/multiplying/dividing a fraction.

11. Omitting units or incorrectly converting units.

I think I'm going to run this list of common mistakes off and give each student a copy so they can use it to double check their steps. Of course, I'll have to model its use but if I use it regularly, perhaps they will choose to use their list.

It is also suggested that the teacher change the way they identify mistakes for students. Rather than saying "You made a mistake", say "I'm glad you made the mistake, it means you are thinking about the problem and you can learn from it." I tend to let the student know they missed a step when solving it, so go back and check to see if they can tell where they missed the step.

In addition it is good for the teacher to make a mistake, correct it, and let the students know what the mistake was and why they did it. It shows that teachers are not infallible. Teachers are human. Too often students are under the mistaken impression that math teachers are extremely smart, like Einstein. Its important to show them we are human. Make it normal to look at mistakes so they are no longer something to be feared but celebrated.

When a student makes a mistake, it is important to correct it but also to understand why the mistake was made. By correcting the error and knowing why it was made, it gives the student a personal sense of success. Furthermore, the type of the mistake provides an assessment for the teacher. The mistakes let the teacher know, what has not been mastered yet.

In a sense, this is something that should be started in elementary but it isn't always so it is necessary to work with students in high school.

Let me know what you think. I love to hear from my readers. Have a good day.

It seems that once the student has completed a problem, their mind shuts the door on it and moves on because they are finished with it and don't need to check it.

One suggestion I ran across is to have a poster in the classroom for the top 11 errors made in math calculations hung somewhere in the room so they can check it before they move on.

1. Did not distribute the outside term to both terms inside the parenthesis. This includes not distributing the negative sign with the number.

2. Multiplying by 2 instead of squaring. In other words they multiply by the exponent, instead of applying the power.

3. Adding instead of subtracting or vice versa.

4. Adding instead of multiplying or vice versa.

5. Misplacing or loosing a decimal.

6. Making a rounding error.

7. Forgetting to carry a number or to borrow.

8. Forgetting to change the inequality sign when dividing or multiplying by a negative.

9. Making a mistake when cross multiplying ratios.

10. Making a mistake when adding/subtracting/multiplying/dividing a fraction.

11. Omitting units or incorrectly converting units.

I think I'm going to run this list of common mistakes off and give each student a copy so they can use it to double check their steps. Of course, I'll have to model its use but if I use it regularly, perhaps they will choose to use their list.

It is also suggested that the teacher change the way they identify mistakes for students. Rather than saying "You made a mistake", say "I'm glad you made the mistake, it means you are thinking about the problem and you can learn from it." I tend to let the student know they missed a step when solving it, so go back and check to see if they can tell where they missed the step.

In addition it is good for the teacher to make a mistake, correct it, and let the students know what the mistake was and why they did it. It shows that teachers are not infallible. Teachers are human. Too often students are under the mistaken impression that math teachers are extremely smart, like Einstein. Its important to show them we are human. Make it normal to look at mistakes so they are no longer something to be feared but celebrated.

When a student makes a mistake, it is important to correct it but also to understand why the mistake was made. By correcting the error and knowing why it was made, it gives the student a personal sense of success. Furthermore, the type of the mistake provides an assessment for the teacher. The mistakes let the teacher know, what has not been mastered yet.

In a sense, this is something that should be started in elementary but it isn't always so it is necessary to work with students in high school.

Let me know what you think. I love to hear from my readers. Have a good day.

## Tuesday, November 14, 2017

### Right Answer?

Too many of my students entering high school are more concerned with getting the right answer. The ones I've had for a while are moving from right answer to did they do the problem correctly. Yes, they want the right answer but they are beginning to look at the whole process rather than only the solution.

The same group believes as long as they get the correct answer, its fine, even if the calculations contained an error. Their answer is usually "So what! I got it right." Even if the calculations is correct but they messed up on one thing in the process but still got the right answer, they still say the same thing. It doesn't matter if it won't work for any other numbers, they don't care because they got the right answer for the problem.

I think its a mind set they get into in elementary school when the problem is right or wrong. As far as I know, most of the elementary school teachers do not take time to teach students to find their mistakes. They focus on teaching process and getting the correct answer.

I've heard of teachers moving away from numerical grades into using a rubric based grading system so its not like 67 percent but rather they are not quite proficient in the topic. It actually sounds more realistic since it eliminates the "How can I bring my grade up?" question. The last test I gave, many students got upset because I said they could make corrections but the corrections themselves would not raise their test grade. Making corrections for the test only allows the students an opportunity to retake a similar test. They don't want that.

I've spoken with the English teacher who said students only want to write one draft, the final draft, before turning it in. They don't understand that in both math and English, it can take multiple tries to get a finished product that finished and ready to be read.

Because they are so focused on the correct answer, they are unable to take what they just did and do the next problem without asking "What do I do?" or "How do I solve this one." Its as if they are totally separate from the problem. I do get students who manage to put it all together to the point they can do all the problems and can help other students learn it. But too many never reach the point.

Then if you ask them how they got the answer, they give you the "I did the work." answer. I realize their is a move to explain how they got the answer but when I was in school, we were told as long as we showed the math we used to arrive at the answer, that explained it all. If I'd been asked to explain how I got the answer, I would have done it by explaining my work at each step.

Right now, I just work on getting the students to look at the whole problem rather than the answer because its easy to make a calculation error such as 3 x 2 = 5 and get the wrong answer while having completed the process correctly.

Is there an answer? I don't know because by the time they get to me in high school, they are convinced getting the right answer is the only thing math focuses on. Let me know what you think.

The same group believes as long as they get the correct answer, its fine, even if the calculations contained an error. Their answer is usually "So what! I got it right." Even if the calculations is correct but they messed up on one thing in the process but still got the right answer, they still say the same thing. It doesn't matter if it won't work for any other numbers, they don't care because they got the right answer for the problem.

I think its a mind set they get into in elementary school when the problem is right or wrong. As far as I know, most of the elementary school teachers do not take time to teach students to find their mistakes. They focus on teaching process and getting the correct answer.

I've heard of teachers moving away from numerical grades into using a rubric based grading system so its not like 67 percent but rather they are not quite proficient in the topic. It actually sounds more realistic since it eliminates the "How can I bring my grade up?" question. The last test I gave, many students got upset because I said they could make corrections but the corrections themselves would not raise their test grade. Making corrections for the test only allows the students an opportunity to retake a similar test. They don't want that.

I've spoken with the English teacher who said students only want to write one draft, the final draft, before turning it in. They don't understand that in both math and English, it can take multiple tries to get a finished product that finished and ready to be read.

Because they are so focused on the correct answer, they are unable to take what they just did and do the next problem without asking "What do I do?" or "How do I solve this one." Its as if they are totally separate from the problem. I do get students who manage to put it all together to the point they can do all the problems and can help other students learn it. But too many never reach the point.

Then if you ask them how they got the answer, they give you the "I did the work." answer. I realize their is a move to explain how they got the answer but when I was in school, we were told as long as we showed the math we used to arrive at the answer, that explained it all. If I'd been asked to explain how I got the answer, I would have done it by explaining my work at each step.

Right now, I just work on getting the students to look at the whole problem rather than the answer because its easy to make a calculation error such as 3 x 2 = 5 and get the wrong answer while having completed the process correctly.

Is there an answer? I don't know because by the time they get to me in high school, they are convinced getting the right answer is the only thing math focuses on. Let me know what you think.

## Monday, November 13, 2017

### Comparing costs.

Every math textbook seems to have a lesson or two on unit costs. Unfortunately, the prices given, even in new textbooks, do not reflect realistic prices where I live in the bush of Alaska.

A Tino's pizza costs around $6 to $8 each. A pizza you can get at one of the grocery chains runs $21.00 for a $8.00 one on sale in town. If I use the problems in the book, my students think something is off and don't relate to them.

Anything frozen that is shipped to the village has the added cost of air freight. This means I might be able to get a half gallon of ice cream for $5.00 in town but by the time its shipped out here, it costs $11 to $13.00. Quite a mark-up.

Then there is the difference in prices locally. For instance, a can of soda runs about $1.25 at the store but various groups buy a 12 pack at the store then resell the soda at $2.00 each. It is a matter of convenience because they do not have to leave the building, drive to the store, and buy it. I have no idea what a soda sells for in the lower 48 because I don't usually buy it.

So in order to give students a real idea of cost comparisons, I have to use local prices. There are five ways I can have students compare prices.

1. Comparing different sizes of the same item to see if the cost per unit is consistent, or which one is the better buy.

2. Rectangular vs circular pizza by weight, are they the same cost?

3. School prices vs city prices. Our school runs a concession stand that is open during sports activities but not during the day because it sells junk food.

4. Comparing prices between the village and Anchorage so students get a better idea of the markup and cost per unit differences. This is easy to do because two stores in Anchorage do bush orders and have websites listing prices.

5. Compare prices between the two stores in town. The stores are more like convenience stores but sell guns, bullets, freezers, and just about anything else a K-mart might sell. I know that sometimes one store has a better price on eggs, milk, or cheese.

Its hard to teach comparing prices using price/unit cost since the store shelves do not have the tags you would normally find. In a normal store, you might have several different brands or sizes of items but since the stores here are small and limited, they do not carry those tags. This means my students do not have the opportunity to learn to read shelf tags the way most students do.

They do not get to see the tags showing the price/unit cost in different units. I've seen two of the same type of product listed with price/product vs price/ounce which makes it much harder compare. So I have to get creative for this type of comparison in the classroom.

What I do have available is the SpanAlaska Catalogue which allows people to order items in bulk. I can have students look up various items and have them calculate the price/unit cost using the same unit. I could have a friend take pictures, send them so I can show students how the tags normally appear. I have to create the experience for my students so when they go somewhere with the shelf tags, they are educated and capable of using them.

Let me know what you think. I'm interested. Have a great day.

A Tino's pizza costs around $6 to $8 each. A pizza you can get at one of the grocery chains runs $21.00 for a $8.00 one on sale in town. If I use the problems in the book, my students think something is off and don't relate to them.

Anything frozen that is shipped to the village has the added cost of air freight. This means I might be able to get a half gallon of ice cream for $5.00 in town but by the time its shipped out here, it costs $11 to $13.00. Quite a mark-up.

Then there is the difference in prices locally. For instance, a can of soda runs about $1.25 at the store but various groups buy a 12 pack at the store then resell the soda at $2.00 each. It is a matter of convenience because they do not have to leave the building, drive to the store, and buy it. I have no idea what a soda sells for in the lower 48 because I don't usually buy it.

So in order to give students a real idea of cost comparisons, I have to use local prices. There are five ways I can have students compare prices.

1. Comparing different sizes of the same item to see if the cost per unit is consistent, or which one is the better buy.

2. Rectangular vs circular pizza by weight, are they the same cost?

3. School prices vs city prices. Our school runs a concession stand that is open during sports activities but not during the day because it sells junk food.

4. Comparing prices between the village and Anchorage so students get a better idea of the markup and cost per unit differences. This is easy to do because two stores in Anchorage do bush orders and have websites listing prices.

5. Compare prices between the two stores in town. The stores are more like convenience stores but sell guns, bullets, freezers, and just about anything else a K-mart might sell. I know that sometimes one store has a better price on eggs, milk, or cheese.

Its hard to teach comparing prices using price/unit cost since the store shelves do not have the tags you would normally find. In a normal store, you might have several different brands or sizes of items but since the stores here are small and limited, they do not carry those tags. This means my students do not have the opportunity to learn to read shelf tags the way most students do.

They do not get to see the tags showing the price/unit cost in different units. I've seen two of the same type of product listed with price/product vs price/ounce which makes it much harder compare. So I have to get creative for this type of comparison in the classroom.

What I do have available is the SpanAlaska Catalogue which allows people to order items in bulk. I can have students look up various items and have them calculate the price/unit cost using the same unit. I could have a friend take pictures, send them so I can show students how the tags normally appear. I have to create the experience for my students so when they go somewhere with the shelf tags, they are educated and capable of using them.

Let me know what you think. I'm interested. Have a great day.

## Sunday, November 12, 2017

## Saturday, November 11, 2017

## Friday, November 10, 2017

### Math Girls.

As I checked out graphic novels, I came across a series of novels called "Math Girls". This series is written by Hiroshi Yuki and the first book was originally published in Japan in 2007.

Since then, the series has sold over 100,000 copies in Japan alone. Beginning in 2011, these novels have been translated into English and now can be bought here in the United States.

The three main characters are the Narrator who is the protagonist and everything is told from his viewpoint, Miruka who is a second year high school student in the same home room as the Narrator, and Tetra a first year high school student who went to the same middle school as the Narrator.In addition, Miruka lives breathes and talks math while Tetra has a serious case of math anxiety.

As far as I can tell, there have been seven volumes released on the Math Girls. The first volume titled Math Girls appears to be the first general introductory volume where we become acquainted. The next three cover equations and graphs, integers, and trig. The next two cover Fermat's Last Theorem and Godel's Incompleteness Theorems. The final one is a volume 2 of the Math Girls Manga.

The books for the most part are made up of conversations between the narrator, Miruka and Tetra along with a supporting cast. You see the dialog, with the mathematical information and proofs sprinkled throughout the story.

Only two books are done in the actual manga style but only one is available in this country. The rest are only available in paperback rather than ebook format. I wouldn't mind having a set in my classroom just so students can see that math can be appealing in a written form that is not a picture book.

Check the books out at Amazon if you'd like to see them in more detail. Let me know what you think. I love hearing from people.

Since then, the series has sold over 100,000 copies in Japan alone. Beginning in 2011, these novels have been translated into English and now can be bought here in the United States.

The three main characters are the Narrator who is the protagonist and everything is told from his viewpoint, Miruka who is a second year high school student in the same home room as the Narrator, and Tetra a first year high school student who went to the same middle school as the Narrator.In addition, Miruka lives breathes and talks math while Tetra has a serious case of math anxiety.

As far as I can tell, there have been seven volumes released on the Math Girls. The first volume titled Math Girls appears to be the first general introductory volume where we become acquainted. The next three cover equations and graphs, integers, and trig. The next two cover Fermat's Last Theorem and Godel's Incompleteness Theorems. The final one is a volume 2 of the Math Girls Manga.

The books for the most part are made up of conversations between the narrator, Miruka and Tetra along with a supporting cast. You see the dialog, with the mathematical information and proofs sprinkled throughout the story.

Only two books are done in the actual manga style but only one is available in this country. The rest are only available in paperback rather than ebook format. I wouldn't mind having a set in my classroom just so students can see that math can be appealing in a written form that is not a picture book.

Check the books out at Amazon if you'd like to see them in more detail. Let me know what you think. I love hearing from people.

## Thursday, November 9, 2017

### What is Number Fluency?

I am always hearing that students need to develop number fluency but what is meant by that? Number fluency is another way of saying students need to develop number sense.

Students need to develop number fluency in conjunction with conceptual understanding and computational fluency. In other words, they need to understand the concept and be about to perform calculations fluidly.

According to one definition, number fluency means a student can compose and decompose numbers in a variety of ways, is able to see patterns in numbers, knows their basic mathematical facts fluently, is able to work fluently and quickly with numbers to solve problems, and is able to work well with place value and with numbers.

People often wonder why number fluency is important. Number fluency is the bridge between recognizing numbers and understanding how to solve problems. A student with number fluency is able to look at the problem rather than focus on basic facts. They are able to make connections with prior knowledge more easily than someone who has to focus on their calculations.

Too many of my students arrive in high school still skip counting on their fingers rather than knowing their multiplication by heart. I realize there are calculators out there but if a student has not developed a sense of what the answer should be, they do not know if their answer is even close. Too many of my students accept the answer from a calculator as the correct value rather than taking time to ask themselves if it is a reasonable answer.

Since almost every student has a mobile device with a calculator, they want to use it rather than trying to remember their multiplication and division facts. If I allow the use of a calculator, I insist that two students work together and each one run the numbers. This way if they disagree, they can try to figure out who put the numbers in incorrectly. This way, I hope it helps them build a sense of the way numbers work.

When it comes to fractions, the first thing they want to do is change the fractions into decimals because they find decimals so much more comfortable to work with. Unfortunately, there are certain fractions that when changed into decimals become repeating numbers and rounding the value does not help.

Once thing I love to do is to have students analyze problems during warm-up to determine if they are correct or incorrect. If they are incorrect, students have to determine what the error is and when it occurred during the calculation. This makes students slow down and think so at least they are developing more of a number sense than what they arrive with in high school.

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

Students need to develop number fluency in conjunction with conceptual understanding and computational fluency. In other words, they need to understand the concept and be about to perform calculations fluidly.

According to one definition, number fluency means a student can compose and decompose numbers in a variety of ways, is able to see patterns in numbers, knows their basic mathematical facts fluently, is able to work fluently and quickly with numbers to solve problems, and is able to work well with place value and with numbers.

People often wonder why number fluency is important. Number fluency is the bridge between recognizing numbers and understanding how to solve problems. A student with number fluency is able to look at the problem rather than focus on basic facts. They are able to make connections with prior knowledge more easily than someone who has to focus on their calculations.

Too many of my students arrive in high school still skip counting on their fingers rather than knowing their multiplication by heart. I realize there are calculators out there but if a student has not developed a sense of what the answer should be, they do not know if their answer is even close. Too many of my students accept the answer from a calculator as the correct value rather than taking time to ask themselves if it is a reasonable answer.

Since almost every student has a mobile device with a calculator, they want to use it rather than trying to remember their multiplication and division facts. If I allow the use of a calculator, I insist that two students work together and each one run the numbers. This way if they disagree, they can try to figure out who put the numbers in incorrectly. This way, I hope it helps them build a sense of the way numbers work.

When it comes to fractions, the first thing they want to do is change the fractions into decimals because they find decimals so much more comfortable to work with. Unfortunately, there are certain fractions that when changed into decimals become repeating numbers and rounding the value does not help.

Once thing I love to do is to have students analyze problems during warm-up to determine if they are correct or incorrect. If they are incorrect, students have to determine what the error is and when it occurred during the calculation. This makes students slow down and think so at least they are developing more of a number sense than what they arrive with in high school.

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

## Wednesday, November 8, 2017

### Adding a Dimension To Fractions.

I am teaching a pre-algebra class this year. I've discovered most of them struggle when adding or subtracting integers. The see the - sign as subtraction rather than a negative number.

I always spend the first semester building their skills before introducing the algebraic element. This year, I am going to do something a bit different.

Instead of teaching fractions using only positive quantities, I want the students to learn fractions are not always positive.

If I find a piece of material, a remnant, that is 1/4 inch short of the length I need, that would indicate a negative value. On the other hand, if the material is 1/4th of a foot over, that would be a positive value.

My students entered high school with certain ideas such as you cannot subtract a larger value from a smaller value so you have a negative result. Like if you write a check for more than you have in your checking account. They also see -4 -6 and do not recognize it as -4 + -6. Even after spending two months on it, they still struggle. I've used chips, number lines, everything I can think of and they still struggle.

I already know they are going to struggle when I write 5 1/4 +(- 1 1/3) instead of 5 1/4 - 1 1/3. I suspect even having them draw pictures and using number lines when they begin working with simple fractions, they will still struggle.

When we start the topic, I plan to have them go onto the internet to find ways in which fractions are used in real life. They'll have to use their own words to describe each situation and provide a picture to illustrate the use. Too often, they do not connect what they learn about fractions in school with their use in real life.

Once this activity is out of the way, I plan to use some activities from Texas Instruments with a bit of modification for my students. The activities range from the general question of "What is a fraction?" to discovering that fractions are equivalent if they are found at the same place on a number line, to mixed numbers. There are 15 different activities in this unit.

When it comes time to discuss common denominators, I've found graph paper is wonderful for creating models designed to show students why any fraction must have the same denominator to combine. Years ago, one of my students admitted they didn't know the boxes had to be subdivided into equal parts.

I also have a couple of games on my ipads for students to play so they can practice using fractions in a more fun way. Towards the end of the unit, I plan to break the students up into groups to create a game using fractions. Once the games are completed, I'll have other groups test the games based on a rubric.

I hope they have an easier time learning this topic than they did learning integers. Let me know what you think. I'd love to hear.

I always spend the first semester building their skills before introducing the algebraic element. This year, I am going to do something a bit different.

Instead of teaching fractions using only positive quantities, I want the students to learn fractions are not always positive.

If I find a piece of material, a remnant, that is 1/4 inch short of the length I need, that would indicate a negative value. On the other hand, if the material is 1/4th of a foot over, that would be a positive value.

My students entered high school with certain ideas such as you cannot subtract a larger value from a smaller value so you have a negative result. Like if you write a check for more than you have in your checking account. They also see -4 -6 and do not recognize it as -4 + -6. Even after spending two months on it, they still struggle. I've used chips, number lines, everything I can think of and they still struggle.

I already know they are going to struggle when I write 5 1/4 +(- 1 1/3) instead of 5 1/4 - 1 1/3. I suspect even having them draw pictures and using number lines when they begin working with simple fractions, they will still struggle.

When we start the topic, I plan to have them go onto the internet to find ways in which fractions are used in real life. They'll have to use their own words to describe each situation and provide a picture to illustrate the use. Too often, they do not connect what they learn about fractions in school with their use in real life.

Once this activity is out of the way, I plan to use some activities from Texas Instruments with a bit of modification for my students. The activities range from the general question of "What is a fraction?" to discovering that fractions are equivalent if they are found at the same place on a number line, to mixed numbers. There are 15 different activities in this unit.

When it comes time to discuss common denominators, I've found graph paper is wonderful for creating models designed to show students why any fraction must have the same denominator to combine. Years ago, one of my students admitted they didn't know the boxes had to be subdivided into equal parts.

I also have a couple of games on my ipads for students to play so they can practice using fractions in a more fun way. Towards the end of the unit, I plan to break the students up into groups to create a game using fractions. Once the games are completed, I'll have other groups test the games based on a rubric.

I hope they have an easier time learning this topic than they did learning integers. Let me know what you think. I'd love to hear.

## Tuesday, November 7, 2017

### Math and Exercise

There is now research indicating that increasing physical activity throughout the day can increase both reading and math scores. It makes sense to me because when I have to sit through a professional development day, I get tired and sluggish.

It appears that regular aerobic exercise in children helps improve their ability to do math. It has been theorized that the aerobic exercise results in a thinner layer of grey matter which is believed to improve cognitive control and working memory. Both of which are important skills in doing math.

It is now being suggested that some sort of physical activity occur in the classroom during the lesson or just before to improve brain function. Even just having them get up and walk around just before presenting an important activity helps increase understanding.

A study from the University of Copenhagen indicates that if the physical activity is integrated into the part of the lesson focused on learning the material, student understanding increases and student scores go up. The physical activity included a whole body element such as drawing shapes with the whole body or using groups of students to do addition or subtraction problems.

It is much harder to create physical activities in the secondary math classroom but at least one high school has managed it by having students take gym first period before math. The gym taught includes things like square dancing which requires the complicated movements designed to stimulate thinking.

In addition, if a math teacher notices a student starting to zone out in class, the teacher implements a short burst of physical activity to get the brain working again. This school has seen its reading scores and math scores increase significantly.

Furthermore, regular physical activity is great for improving behavior, concentration, and physical shape. It can also decrease a person's anxiety prior to a major exam. Research has discovered there is an immediate improvement in concentration as soon as they begin regular exercise. Over time, the ability of the brain to process mathematics improves.

I found this to be quite interesting. I think I might need to include a bit of physical activity in the middle of my classes to increase brain power. Students take 7 classes per day with only a four minute break between and one 30 min lunch period. Not really enough time for the brain to get the stimulation it needs.

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

It appears that regular aerobic exercise in children helps improve their ability to do math. It has been theorized that the aerobic exercise results in a thinner layer of grey matter which is believed to improve cognitive control and working memory. Both of which are important skills in doing math.

It is now being suggested that some sort of physical activity occur in the classroom during the lesson or just before to improve brain function. Even just having them get up and walk around just before presenting an important activity helps increase understanding.

A study from the University of Copenhagen indicates that if the physical activity is integrated into the part of the lesson focused on learning the material, student understanding increases and student scores go up. The physical activity included a whole body element such as drawing shapes with the whole body or using groups of students to do addition or subtraction problems.

It is much harder to create physical activities in the secondary math classroom but at least one high school has managed it by having students take gym first period before math. The gym taught includes things like square dancing which requires the complicated movements designed to stimulate thinking.

In addition, if a math teacher notices a student starting to zone out in class, the teacher implements a short burst of physical activity to get the brain working again. This school has seen its reading scores and math scores increase significantly.

Furthermore, regular physical activity is great for improving behavior, concentration, and physical shape. It can also decrease a person's anxiety prior to a major exam. Research has discovered there is an immediate improvement in concentration as soon as they begin regular exercise. Over time, the ability of the brain to process mathematics improves.

I found this to be quite interesting. I think I might need to include a bit of physical activity in the middle of my classes to increase brain power. Students take 7 classes per day with only a four minute break between and one 30 min lunch period. Not really enough time for the brain to get the stimulation it needs.

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

## Monday, November 6, 2017

### Real life inequalities

Its hard for people to identify situations involving inequalities in real life because they are not neatly laid out in the same manner as found in textbooks.

Inequalities are all around us but most teachers spend time teaching students to create problems from written word problems or learning to solve problems without spending much time looking at how these problems appear in real life.

So I'm taking time to look at some everyday situations which are inequalities so perhaps we can expose our students to more problems as they look in real life, not in the text book.

1. Trucking companies are very aware of inequalities because they have to deal with weight limits set by bridges. Most bridges have an upper limit for the maximum weight of a truck, its cab, and its cargo so companies must be aware of how much cargo they can load the truck with since the weight of the cab and trailer is static.

An example might be the truck has to cross several bridges with a maximum weight of 65,000 lbs. The cab is 21,000 pounds while the trailer's empty weight is 19,000 pounds. This means the cab and trailer weigh 40,000 pounds so the truck cannot have more than 25,000 pounds of cargo. The inequality is less than or equal to 25,000 pounds.

Airplanes and trains also have to watch their weight due to fuel consumption based on number of passengers, temperature a few other things.

2. Another inequality problem is when we want to buy something like sneakers and there are several models we like but they each cost a different price. When we need to know how many hours we need to work to afford them, we look at the cost of the cheapest pair to find a minimum.

An example might be that we've found three pair of basketball shoes we love priced $140, $143, $148. We've already saved $40 towards a pair so we'll need to determine how many hours we need to work to afford the cheapest pair. This is a greater than or equal to type problem because we need to know the minimum amount we need to earn.

3. I worked a job one time where I got paid so much per hour but if I sold at least $250 worth of product each week, I would start making more per week. I received a set amount of money for each item I sold plus the base wage. I could easily determine how many items I needed to sell to reach a certain salary.

4. When you take the elevator, there is a maximum weight associated with the number of people the elevator can safely handle. This type of problem is a less than or equal to because the weight cannot be more than a certain number. In addition, they use an average weight for people to determine the maximum number the elevator can handle. This means you can have more if they all weigh less then the average or fewer if they weigh more. The next time you ride an elevator check it out.

5. Sometimes you find inequalities when dealing with the number of letters placed on a t-shirt or on a piece of jewelry. For instance, you are charged $0.10 per letter and you have a budget of $5.00, how many letters can you afford and what are some of the things you can have engraved on the jewelry?

These are just a few situations in real life which use inequalities. I like them because I relate to the situations and so do most everyone else. Let me know what you think. Have a great day.

Inequalities are all around us but most teachers spend time teaching students to create problems from written word problems or learning to solve problems without spending much time looking at how these problems appear in real life.

So I'm taking time to look at some everyday situations which are inequalities so perhaps we can expose our students to more problems as they look in real life, not in the text book.

1. Trucking companies are very aware of inequalities because they have to deal with weight limits set by bridges. Most bridges have an upper limit for the maximum weight of a truck, its cab, and its cargo so companies must be aware of how much cargo they can load the truck with since the weight of the cab and trailer is static.

An example might be the truck has to cross several bridges with a maximum weight of 65,000 lbs. The cab is 21,000 pounds while the trailer's empty weight is 19,000 pounds. This means the cab and trailer weigh 40,000 pounds so the truck cannot have more than 25,000 pounds of cargo. The inequality is less than or equal to 25,000 pounds.

Airplanes and trains also have to watch their weight due to fuel consumption based on number of passengers, temperature a few other things.

2. Another inequality problem is when we want to buy something like sneakers and there are several models we like but they each cost a different price. When we need to know how many hours we need to work to afford them, we look at the cost of the cheapest pair to find a minimum.

An example might be that we've found three pair of basketball shoes we love priced $140, $143, $148. We've already saved $40 towards a pair so we'll need to determine how many hours we need to work to afford the cheapest pair. This is a greater than or equal to type problem because we need to know the minimum amount we need to earn.

3. I worked a job one time where I got paid so much per hour but if I sold at least $250 worth of product each week, I would start making more per week. I received a set amount of money for each item I sold plus the base wage. I could easily determine how many items I needed to sell to reach a certain salary.

4. When you take the elevator, there is a maximum weight associated with the number of people the elevator can safely handle. This type of problem is a less than or equal to because the weight cannot be more than a certain number. In addition, they use an average weight for people to determine the maximum number the elevator can handle. This means you can have more if they all weigh less then the average or fewer if they weigh more. The next time you ride an elevator check it out.

5. Sometimes you find inequalities when dealing with the number of letters placed on a t-shirt or on a piece of jewelry. For instance, you are charged $0.10 per letter and you have a budget of $5.00, how many letters can you afford and what are some of the things you can have engraved on the jewelry?

These are just a few situations in real life which use inequalities. I like them because I relate to the situations and so do most everyone else. Let me know what you think. Have a great day.

## Sunday, November 5, 2017

## Saturday, November 4, 2017

## Friday, November 3, 2017

### Proportional Reasoning.

Proportional reasoning is the ability to compare two things using multiplicative thinking. Something like when we have students use proportional reasoning to identify similar triangles, or solving for the unknown. Sometimes we find problems in the book for enlarging photos or scale models.

I've seldom seen it applied to a real life situation until I came across an article applying it to social justice. There are several types of problems which fall under the general topic of social justice.

1. Calculating population growth and the associated crime rates. One way to handle this is to look at the population growth for local populations so students can determine when they need to find differences and when they need ratios. They also need to determine when starting population differences make a difference and when they do not such as in building a fixed number of houses or creating a new electrical grid.

Population growth is important. I have some cousins who live outside of Washington, D.C. in Virginia. The area underwent tremendous growth because people could still find a great deal in the cost of housing but traffic became extremely bad when the area could not keep up with the number of cars on the road.

When you start to compare crime rates in the same areas and look only at violent crimes, students can look at ratios and percentages to determine which location is safer based on that criteria.

2. Look at the representation in Congress versus the population of racial composition by state or over all. If you wanted, you could take a look at the representation in the state government versus the racial composition of that state.

3. Look at historical immigration rates from 1820 to 2000 to see how rates have changed over time. You could even look at recent rates to see where the majority of immigrants come from.

4. Look at the pay people who work in sweat shops make versus the cost the item sells for in stores in the United States. Calculate the daily, monthly, and yearly rates earned by these workers.

This google site has a wonderful list of places to find additional lessons connecting proportional reasoning with social justice. In addition, many of the lessons have students applying mathematics to social topics.

In addition, the October 2015 issue of Mathematics teaching in the Middle School magazine has great information for turning general textbook problems into more personalized problems students can relate to better.

Let me know what you think. I look forward to hearing from people.

I've seldom seen it applied to a real life situation until I came across an article applying it to social justice. There are several types of problems which fall under the general topic of social justice.

1. Calculating population growth and the associated crime rates. One way to handle this is to look at the population growth for local populations so students can determine when they need to find differences and when they need ratios. They also need to determine when starting population differences make a difference and when they do not such as in building a fixed number of houses or creating a new electrical grid.

Population growth is important. I have some cousins who live outside of Washington, D.C. in Virginia. The area underwent tremendous growth because people could still find a great deal in the cost of housing but traffic became extremely bad when the area could not keep up with the number of cars on the road.

When you start to compare crime rates in the same areas and look only at violent crimes, students can look at ratios and percentages to determine which location is safer based on that criteria.

2. Look at the representation in Congress versus the population of racial composition by state or over all. If you wanted, you could take a look at the representation in the state government versus the racial composition of that state.

3. Look at historical immigration rates from 1820 to 2000 to see how rates have changed over time. You could even look at recent rates to see where the majority of immigrants come from.

4. Look at the pay people who work in sweat shops make versus the cost the item sells for in stores in the United States. Calculate the daily, monthly, and yearly rates earned by these workers.

This google site has a wonderful list of places to find additional lessons connecting proportional reasoning with social justice. In addition, many of the lessons have students applying mathematics to social topics.

In addition, the October 2015 issue of Mathematics teaching in the Middle School magazine has great information for turning general textbook problems into more personalized problems students can relate to better.

Let me know what you think. I look forward to hearing from people.

## Thursday, November 2, 2017

### Designing Sports Bags

If you every look at crowd sourcing site on line, you'll find an assortment of groups who market the "perfect bag". Yes, I've invested in a few of which one has gone into production and been delivered while the other one will soon arrive.

I invested in a backpack that allowed me to spend a couple weeks in Europe with only a small roller bag as my check through. It was great. I spotted a task which would be perfect in the Geometry classroom based on shapes and nets.

Why no assign students the task of designing a sports bag or gym bag to certain specifications just like designers do in the real world? The teacher is the manager who assigns the students to design and make a patterns for the sewing room to make the object.

Criteria could include:

1. The finished bag is 55 cm long

2. The circular ends must have a diameter of 22 cm.

3. The body is made from a single piece of fabric.

4. The ends must be made from the same fabric.

5. Make sure there is an extra 1.5 cm added to all sides for the seam.

6. The fabric is 1 meter wide.

The student needs to create a pattern with the exact shapes needed. In addition, the student needs to determine the smallest amount of fabric needed to make the bag. This means they need to create a cutting guide, just like the ones found in patterns.

A perfect place to insert this type of activity in the geometry class is when studying the nets needed to make three dimensional shapes such as cylinders, pyramids, etc. I love finding any application I can for real life applications. In the past, I've had them just create the nets for various shapes but this activity takes it a step further.

There are a couple sites which have wonderful lesson plans for this activity. One is at the Mathematics Assessment Project, which has everything needed to run this in your classroom. If you prefer, you could have students design a backpack (rectangular prism) or other type of bag.

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

I invested in a backpack that allowed me to spend a couple weeks in Europe with only a small roller bag as my check through. It was great. I spotted a task which would be perfect in the Geometry classroom based on shapes and nets.

Why no assign students the task of designing a sports bag or gym bag to certain specifications just like designers do in the real world? The teacher is the manager who assigns the students to design and make a patterns for the sewing room to make the object.

Criteria could include:

1. The finished bag is 55 cm long

2. The circular ends must have a diameter of 22 cm.

3. The body is made from a single piece of fabric.

4. The ends must be made from the same fabric.

5. Make sure there is an extra 1.5 cm added to all sides for the seam.

6. The fabric is 1 meter wide.

The student needs to create a pattern with the exact shapes needed. In addition, the student needs to determine the smallest amount of fabric needed to make the bag. This means they need to create a cutting guide, just like the ones found in patterns.

A perfect place to insert this type of activity in the geometry class is when studying the nets needed to make three dimensional shapes such as cylinders, pyramids, etc. I love finding any application I can for real life applications. In the past, I've had them just create the nets for various shapes but this activity takes it a step further.

There are a couple sites which have wonderful lesson plans for this activity. One is at the Mathematics Assessment Project, which has everything needed to run this in your classroom. If you prefer, you could have students design a backpack (rectangular prism) or other type of bag.

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

## Wednesday, November 1, 2017

### The Moon and Math.

We are in the time of the year in Alaska, when we will often see the moon for 24 hours or longer at a time. There are times when it does not set but circles around the sky. It is an awesome sight.

The moon and its phases offers some great math opportunities for the classroom in geometry and in measuring time. Two different aspects of something we see all our lives.

Lets start with the relationship between the time of day and the visibility of the moon based on its phase. It is important to remember the moonlight we see is actually sunlight bouncing off the moon. Consequently, the phase of the moon is determined by the relationship between the moon, the earth and the sun.

So the geometry involved is as follows. When the moon is between the earth and the sun, we observe it as a new moon because we are looking at the moon without illumination. Two weeks later when the moon is on the other side of the earth from the sun, the moon is fully illuminated and we see it as the full moon. The two quarter moons occur in between when the moon is half lit. When all this is happening, the moon is rising as the sun is setting during the full moon.

Add in an activity that uses a spreadsheet to determine the next full moon. The moon's orbit is 29.5 days long so its a matter of adding that amount to the current time of moon rise. However, if you want to make it a bit more interesting follow the link to get the in depth equations to create a wonderful sine wave visualization of the moons cycle. The equation even goes so far as to include the percent of the phases of the moon. The author has them adjust the equation to make it a bit more accurate. See if you could coordinate this activity with the science class when they are studying phases of the moon.

Of course NASA has a 128 page book filled with various science and mathematical activities for grades 3 to 12 on lunar math. It includes a list of math topics covered in this book. I took a look at some of the activities for younger students and with a bit of adjustment, they could be used in the high school.

This lesson geared for middle school seems to tie the geometry of moon phases and spread sheet into one nice long lesson. It has everything needed and all the links for supplemental materials.

Finally, there is a four lesson unit in the NCTM middle school magazine which integrates science, math, and literacy on using moon phases to measure time. Lesson one is focused on lunar phases and how they work. The second lesson focuses on what make a good standard unit because we measure things in standard units such as feet or meters. The third lesson focuses on the Hopi because their astronomers did a good job of marking time. This lesson includes the lunar cycles based on the Hopi names for the moons. The final lesson looks at the ways different cultures describe time.

Have a good day and let me know what you think. I love getting feedback.

The moon and its phases offers some great math opportunities for the classroom in geometry and in measuring time. Two different aspects of something we see all our lives.

Lets start with the relationship between the time of day and the visibility of the moon based on its phase. It is important to remember the moonlight we see is actually sunlight bouncing off the moon. Consequently, the phase of the moon is determined by the relationship between the moon, the earth and the sun.

So the geometry involved is as follows. When the moon is between the earth and the sun, we observe it as a new moon because we are looking at the moon without illumination. Two weeks later when the moon is on the other side of the earth from the sun, the moon is fully illuminated and we see it as the full moon. The two quarter moons occur in between when the moon is half lit. When all this is happening, the moon is rising as the sun is setting during the full moon.

Add in an activity that uses a spreadsheet to determine the next full moon. The moon's orbit is 29.5 days long so its a matter of adding that amount to the current time of moon rise. However, if you want to make it a bit more interesting follow the link to get the in depth equations to create a wonderful sine wave visualization of the moons cycle. The equation even goes so far as to include the percent of the phases of the moon. The author has them adjust the equation to make it a bit more accurate. See if you could coordinate this activity with the science class when they are studying phases of the moon.

Of course NASA has a 128 page book filled with various science and mathematical activities for grades 3 to 12 on lunar math. It includes a list of math topics covered in this book. I took a look at some of the activities for younger students and with a bit of adjustment, they could be used in the high school.

This lesson geared for middle school seems to tie the geometry of moon phases and spread sheet into one nice long lesson. It has everything needed and all the links for supplemental materials.

Finally, there is a four lesson unit in the NCTM middle school magazine which integrates science, math, and literacy on using moon phases to measure time. Lesson one is focused on lunar phases and how they work. The second lesson focuses on what make a good standard unit because we measure things in standard units such as feet or meters. The third lesson focuses on the Hopi because their astronomers did a good job of marking time. This lesson includes the lunar cycles based on the Hopi names for the moons. The final lesson looks at the ways different cultures describe time.

Have a good day and let me know what you think. I love getting feedback.

## Tuesday, October 31, 2017

### Halloween Math.

Happy Halloween to everyone. Most of the time when I've gone looking for Halloween based math, the ones I find are all geared for elementary school.

I want problems geared for a higher level and which are more interesting than "Jane has 8 pumpkins. She gives 3 to John. How many pumpkins does she have left?" My high schoolers are not interested in those.

So after some searching I found some lovely problems geared more for middle school and high school students. Problems that are a bit more interesting.

1. The worlds largest pumpkin weighs 2222 pounds in 2017. The average pumpkin weighs 12 pounds, how many pumpkins does the largest pumpkin equal.

2. The record for the distance a pumpkin has been thrown in the chuckin contest is 4484 feet. What percentage of a mile is that?

3. The first Halloween parade happened in 1920. How many months till it celebrates its centenial?

4. Buy a couple packages of pumpkins from the seasonal section to create a matching game. You might create matching games where one student multiplies two binomials while the other factors a trinomial. The students do the math then look for who they match. Another possibility is to create equations in pairs so they get the same answer. Another possibility, one card has the equation while the other has the graph to match the equation.

5. The Census bureau has some wonderful data concerning Halloween such as the number of potential Trick or Treaters ages 5 to 14, the number of potential stops based on the number of housing units, number of units with stairs, number of people employed to make the candy, and other wonderful facts.

6. How about an estimation activity to estimate the number of seeds in a pumpkin, estimate the circumference, estimate its weight. After the estimations, have students count the actual number of seeds to determine the closest guess, then make spiced pumpkin seeds to eat in the class. After finding the circumference, calculate the diameter. Finally after giving the weight to the students, have them bake the pumpkin and use the pulp to make pumpkin bread to eat. You can weigh the pumpkin when you buy it so you'll already know that.

7. Have students create a haunted house on a coordinate. They can then give the location of items within the haunted house. They can also find the distance between each part.

8. Students can create graphs containing the name of the candy, the number of calories per serving and its weight in grams. Once done, students need to calculate the number of calories per gram so they can determine the candy with the highest or lowest per unit values.

9. Begin a Pascal's triangle with pumpkins. There should be enough pumpkins on the paper to help students recognize the pattern so they can finish it. It can lead to a great discussion on Pascal's triangles.

Just a few suggestions to make math and Halloween a bit more interesting. Have a great day and let me know what you think. I love to hear.

I want problems geared for a higher level and which are more interesting than "Jane has 8 pumpkins. She gives 3 to John. How many pumpkins does she have left?" My high schoolers are not interested in those.

So after some searching I found some lovely problems geared more for middle school and high school students. Problems that are a bit more interesting.

1. The worlds largest pumpkin weighs 2222 pounds in 2017. The average pumpkin weighs 12 pounds, how many pumpkins does the largest pumpkin equal.

2. The record for the distance a pumpkin has been thrown in the chuckin contest is 4484 feet. What percentage of a mile is that?

3. The first Halloween parade happened in 1920. How many months till it celebrates its centenial?

4. Buy a couple packages of pumpkins from the seasonal section to create a matching game. You might create matching games where one student multiplies two binomials while the other factors a trinomial. The students do the math then look for who they match. Another possibility is to create equations in pairs so they get the same answer. Another possibility, one card has the equation while the other has the graph to match the equation.

5. The Census bureau has some wonderful data concerning Halloween such as the number of potential Trick or Treaters ages 5 to 14, the number of potential stops based on the number of housing units, number of units with stairs, number of people employed to make the candy, and other wonderful facts.

6. How about an estimation activity to estimate the number of seeds in a pumpkin, estimate the circumference, estimate its weight. After the estimations, have students count the actual number of seeds to determine the closest guess, then make spiced pumpkin seeds to eat in the class. After finding the circumference, calculate the diameter. Finally after giving the weight to the students, have them bake the pumpkin and use the pulp to make pumpkin bread to eat. You can weigh the pumpkin when you buy it so you'll already know that.

7. Have students create a haunted house on a coordinate. They can then give the location of items within the haunted house. They can also find the distance between each part.

8. Students can create graphs containing the name of the candy, the number of calories per serving and its weight in grams. Once done, students need to calculate the number of calories per gram so they can determine the candy with the highest or lowest per unit values.

9. Begin a Pascal's triangle with pumpkins. There should be enough pumpkins on the paper to help students recognize the pattern so they can finish it. It can lead to a great discussion on Pascal's triangles.

Just a few suggestions to make math and Halloween a bit more interesting. Have a great day and let me know what you think. I love to hear.

## Monday, October 30, 2017

### Pumpkin Chunkin

It is almost Halloween, the time of pumpkins and crazies who launch pumpkins with huge self built catapults. This three decade old tradition raises funds for various groups and is due to happen the first weekend in November.

This event brings its followers in from all over the place. I have friends who disappear for the weekend so they can go out to Delaware to watch.

There is some great math involved in this event from the parabolic projectile path to the time it takes for the pumpkin to travel across the field, etc. Most people launch their pumpkins using some sort of air cannon to provide the force necessary to launch the object.

Lets start with finding the initial velocity by using the formula x*(k/m)^1/2 with m representing mass, k equals the spring constant, and x is distance in meters. Since the pumpkin is being shot out of an object, you are dealing with projectile motion. It is this motion which creates a wonderful parabolic shape. We can use the projectile motion equation d = (v^2osin(2theta))/g to find distance. Add into this finding the time of the pumpkin's trip by using t = Vo sin(angle)

If they calculate the flight of the pumpkin without air resistance, then . There is vertical and horizontal velocities, the angle of launch, and acceleration out of the cannon. So you end up with the equation Vv= Vo sin(angle) and Vh = Vo cos(angle). A nice practical use for trigonometry.

If you add in drag, the equation changes to drag = 1/2 * items density * speed^2 * area * drag coefficient. Drag is what slows the object down and the area listed in the equation refers to the cross sectional area of the pumpkin which can be found using the equation pi * diamter^2/4.

Lots of cool math involved in shooting off a pumpkin. Yes it is mostly involves physics but its still the math involved when shooting a pumpkin. So many things. Just a fun fact, the pumpkin chuckers are still trying to hurl the pumpkin at least one mile but so far they have managed about 4,450 feet.

If you are interested in more math equations, just check out the internet for one of many articles with all the math equations needed to calculate just about anything to do with this event. I chose just a few of the equations. In addition, there are all sorts of plans to follow to build launchers. Imagine having students build small versions of the launchers designed to launch those mini pumpkins. Imagine having students follow up with the math to prove things.

Let me know what you think! Have a great day. Tomorrow, I'm looking Halloween based math appropriate for use in the middle school or high school.

This event brings its followers in from all over the place. I have friends who disappear for the weekend so they can go out to Delaware to watch.

There is some great math involved in this event from the parabolic projectile path to the time it takes for the pumpkin to travel across the field, etc. Most people launch their pumpkins using some sort of air cannon to provide the force necessary to launch the object.

Lets start with finding the initial velocity by using the formula x*(k/m)^1/2 with m representing mass, k equals the spring constant, and x is distance in meters. Since the pumpkin is being shot out of an object, you are dealing with projectile motion. It is this motion which creates a wonderful parabolic shape. We can use the projectile motion equation d = (v^2osin(2theta))/g to find distance. Add into this finding the time of the pumpkin's trip by using t = Vo sin(angle)

If they calculate the flight of the pumpkin without air resistance, then . There is vertical and horizontal velocities, the angle of launch, and acceleration out of the cannon. So you end up with the equation Vv= Vo sin(angle) and Vh = Vo cos(angle). A nice practical use for trigonometry.

If you add in drag, the equation changes to drag = 1/2 * items density * speed^2 * area * drag coefficient. Drag is what slows the object down and the area listed in the equation refers to the cross sectional area of the pumpkin which can be found using the equation pi * diamter^2/4.

Lots of cool math involved in shooting off a pumpkin. Yes it is mostly involves physics but its still the math involved when shooting a pumpkin. So many things. Just a fun fact, the pumpkin chuckers are still trying to hurl the pumpkin at least one mile but so far they have managed about 4,450 feet.

If you are interested in more math equations, just check out the internet for one of many articles with all the math equations needed to calculate just about anything to do with this event. I chose just a few of the equations. In addition, there are all sorts of plans to follow to build launchers. Imagine having students build small versions of the launchers designed to launch those mini pumpkins. Imagine having students follow up with the math to prove things.

Let me know what you think! Have a great day. Tomorrow, I'm looking Halloween based math appropriate for use in the middle school or high school.

## Sunday, October 29, 2017

## Saturday, October 28, 2017

## Friday, October 27, 2017

### Pattern Blocks in the Secondary Classroom.

My experience with pattern blocks is really next to nothing because I've only ever seen them used in the elementary grades. I knew they could be used to teach parts of the whole but I never realized they could be used at the middle school and high school levels.

I've actually discovered a few ways they can be used in the upper grades. I am thrilled because I have a collection of them from the previous teacher. There are also a couple of apps for sale from the app store so you can use them on the iPad.

The first way to use them, is by having students create semi-regular tessellations using pattern blocks. Imagine having students using squares and blue rhombus. Or using two or more regular shapes to cover the plane in the same polygon order either clockwise or counter clockwise.

Let their creativity loose so they can create repeated patterns within a circular shape or a rectangular shape.

Unfortunately, many students reach middle school or high school without a solid foundation in using fractions. Pattern blocks are a good manipulative for teaching fractions in a way that might be more understandable for students. This lesson found on better lesson is specifically designed to teach fractions to students using pattern blocks. The lesson has an anticapatory set through the end activity.

The math learning center has a nice pdf filled with activities for grades 3 to 5 all designed to teach basic mathematical concepts from symmetry to angles. In addition, Henri Picciotto has a free ebook filled with ideas including several activities which use pattern blocks to learn more about angles, polygons, symmetry, measurement, and similarity. He also includes a presentation for using pattern blocks in the classroom, and a link for pattern block trains for teaching rates of change.

What about using pattern blocks to teach functions? Check out this pdf with an activity for creating functions. The files include both the worksheets and the answers which is nice time saver.

If you want to show mathematics and pattern blocks in a slightly different light, check out this place which equates the pattern block shapes with number of beats on a drum. Its like using the blocks instead of notes for the music. The smallest shape, the triangle, is worth one beat, the rhombus is worth two, etc. It is easy to arrange for the use of drums out here because the drum is the basic instrument for dances. The school owns several.

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

I've actually discovered a few ways they can be used in the upper grades. I am thrilled because I have a collection of them from the previous teacher. There are also a couple of apps for sale from the app store so you can use them on the iPad.

The first way to use them, is by having students create semi-regular tessellations using pattern blocks. Imagine having students using squares and blue rhombus. Or using two or more regular shapes to cover the plane in the same polygon order either clockwise or counter clockwise.

Let their creativity loose so they can create repeated patterns within a circular shape or a rectangular shape.

Unfortunately, many students reach middle school or high school without a solid foundation in using fractions. Pattern blocks are a good manipulative for teaching fractions in a way that might be more understandable for students. This lesson found on better lesson is specifically designed to teach fractions to students using pattern blocks. The lesson has an anticapatory set through the end activity.

The math learning center has a nice pdf filled with activities for grades 3 to 5 all designed to teach basic mathematical concepts from symmetry to angles. In addition, Henri Picciotto has a free ebook filled with ideas including several activities which use pattern blocks to learn more about angles, polygons, symmetry, measurement, and similarity. He also includes a presentation for using pattern blocks in the classroom, and a link for pattern block trains for teaching rates of change.

What about using pattern blocks to teach functions? Check out this pdf with an activity for creating functions. The files include both the worksheets and the answers which is nice time saver.

If you want to show mathematics and pattern blocks in a slightly different light, check out this place which equates the pattern block shapes with number of beats on a drum. Its like using the blocks instead of notes for the music. The smallest shape, the triangle, is worth one beat, the rhombus is worth two, etc. It is easy to arrange for the use of drums out here because the drum is the basic instrument for dances. The school owns several.

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

## Thursday, October 26, 2017

### Math Links

I love exploring the internet because there are so many great sites for math teachers. Every time I research something, I find a great site I can use in my classroom.

The other day, I found Math Links Network, a site filled with over 1500 links for teaching math.

Each link has a picture, the date added, and the URL of the site so you can explore it. In addition, the links are divided into one of six categories: Numeracy, patterns and algebra, data, measurement, space and geometry, and advanced. When you click on the category, there is a list of subtopics at the top of the page letting you know how many links are in each.

If you click on Space and Geometry, you'll see 16 sub topics such as triangles, circles, angles, etc. I clicked on triangles with the 18 links listed. Some of the links are for sites and videos I've never seen before and will provide additional materials for my classes.

There are other offerings by this site for teachers. The Maths Faculty has 213 resources from other teachers, designed for teachers. These resources are divided up into the same 6 categories as the links take the explorer back to many of the same links as the main page but not all of them.

I found a maths kit list of links grouped by type. These categories are way different than the other pages in that they have links to graph paper, calculators, dice, equation editors, videos and other such topics. Every grouping has at least three entries but most have 10 or more. The site with the URL is listed for each link.

Another section offers different types of papers needed in the math classroom. Papers such as coordinate grids, graph paper, Isometric dot paper, and number lines, trig graph paper and parabolic graph paper. Each section has several different types of the paper. This is the place to go when looking for specialty paper rather than having to scour the internet.

The final section is the starters section with 13 different starters, timers and other things to make teaching a bit easier.

I admit, this site has so much on it, that I'll be a long time checking things out, determining which activities will work on the iPad and which won't. I didn't know there was a site like this out there. It appears to be from Australia but the links are worth investigating.

Let me know what you think. I'd love to hear. Have a great day.

The other day, I found Math Links Network, a site filled with over 1500 links for teaching math.

Each link has a picture, the date added, and the URL of the site so you can explore it. In addition, the links are divided into one of six categories: Numeracy, patterns and algebra, data, measurement, space and geometry, and advanced. When you click on the category, there is a list of subtopics at the top of the page letting you know how many links are in each.

If you click on Space and Geometry, you'll see 16 sub topics such as triangles, circles, angles, etc. I clicked on triangles with the 18 links listed. Some of the links are for sites and videos I've never seen before and will provide additional materials for my classes.

There are other offerings by this site for teachers. The Maths Faculty has 213 resources from other teachers, designed for teachers. These resources are divided up into the same 6 categories as the links take the explorer back to many of the same links as the main page but not all of them.

I found a maths kit list of links grouped by type. These categories are way different than the other pages in that they have links to graph paper, calculators, dice, equation editors, videos and other such topics. Every grouping has at least three entries but most have 10 or more. The site with the URL is listed for each link.

Another section offers different types of papers needed in the math classroom. Papers such as coordinate grids, graph paper, Isometric dot paper, and number lines, trig graph paper and parabolic graph paper. Each section has several different types of the paper. This is the place to go when looking for specialty paper rather than having to scour the internet.

The final section is the starters section with 13 different starters, timers and other things to make teaching a bit easier.

I admit, this site has so much on it, that I'll be a long time checking things out, determining which activities will work on the iPad and which won't. I didn't know there was a site like this out there. It appears to be from Australia but the links are worth investigating.

Let me know what you think. I'd love to hear. Have a great day.

## Wednesday, October 25, 2017

### Solve Me Moble Math

Did you ever try to make a mobile in school when you were younger? Do you remember how hard it was to get it to balance properly and when you finally did it, you celebrated. Well, while reading up on emoji math, there was mention of something called Solve me Mobiles.

Solve Me Mobiles math is an online math place developed by the National Science Foundation. It also works on both computers and iPads. In addition, there is a free app available for the iPad if you don't want to use the browser on the iPad.The idea behind this site is when both sides of the mobile are balanced, the values are correct. If the values are not correct, then it is unbalanced.

This site follows the same idea as the emoji math site except it uses basic shapes to represent values. The site is divided into two parts, play and build.

In the play section, the first 60 problems are designed to explore how this mobile algebra site works. Some of the problems have totals while others have you make an educated guess. The second section is the puzzler section with more complex problems such as 3x + 2 = 2x -4 represented by fractions. The final section is the master section has 80 complex problems.The problems are designed to begin with simple problems moving to the more complex problems as a person develops their skills.

The second part is designed to allow anyone to create their own balance puzzles. It allows you to choose the shapes and the values for the shapes so you could actually take an equation and set it up on the balance to find the answers. Its also great for just letting students explore the concept of equality between the two sides of the balance.

For more information on how to use this site in class, there is a short video on this web page on using it to learn the logic of solving systems. The video is just over 2.5 minutes long. The creator of the video states their research indicates Solve me mobiles helps math students transfer knowledge better.

One teacher who uses this site in class as a warm-up, assigns specific problems to do before taking time to lead a conversation on which strategies they used to solve the problems. I love this idea because it helps students develop their ability to communicate mathematical ideas. In addition, it engages students and builds the knowledge they need to use in solving standard algebraic equations.

Another teacher who used this site had students build their own puzzles only after they created them on paper so they knew exactly what they wanted to create and to make sure it works. Do a quick search and you'll find several videos out there on the site.

I love playing with the puzzles myself. I'm think my students are going to have fun playing with this site and the emoji math site. One thing I learned early on, is that if you send them to a web site, its best to put the URL in a QR code to make it easier for students to get to the site.

Let me know what you think. Tomorrow, I'll be sharing a site I found filled with links for those of us who teach math. Have a great day.

Solve Me Mobiles math is an online math place developed by the National Science Foundation. It also works on both computers and iPads. In addition, there is a free app available for the iPad if you don't want to use the browser on the iPad.The idea behind this site is when both sides of the mobile are balanced, the values are correct. If the values are not correct, then it is unbalanced.

This site follows the same idea as the emoji math site except it uses basic shapes to represent values. The site is divided into two parts, play and build.

In the play section, the first 60 problems are designed to explore how this mobile algebra site works. Some of the problems have totals while others have you make an educated guess. The second section is the puzzler section with more complex problems such as 3x + 2 = 2x -4 represented by fractions. The final section is the master section has 80 complex problems.The problems are designed to begin with simple problems moving to the more complex problems as a person develops their skills.

The second part is designed to allow anyone to create their own balance puzzles. It allows you to choose the shapes and the values for the shapes so you could actually take an equation and set it up on the balance to find the answers. Its also great for just letting students explore the concept of equality between the two sides of the balance.

For more information on how to use this site in class, there is a short video on this web page on using it to learn the logic of solving systems. The video is just over 2.5 minutes long. The creator of the video states their research indicates Solve me mobiles helps math students transfer knowledge better.

One teacher who uses this site in class as a warm-up, assigns specific problems to do before taking time to lead a conversation on which strategies they used to solve the problems. I love this idea because it helps students develop their ability to communicate mathematical ideas. In addition, it engages students and builds the knowledge they need to use in solving standard algebraic equations.

Another teacher who used this site had students build their own puzzles only after they created them on paper so they knew exactly what they wanted to create and to make sure it works. Do a quick search and you'll find several videos out there on the site.

I love playing with the puzzles myself. I'm think my students are going to have fun playing with this site and the emoji math site. One thing I learned early on, is that if you send them to a web site, its best to put the URL in a QR code to make it easier for students to get to the site.

Let me know what you think. Tomorrow, I'll be sharing a site I found filled with links for those of us who teach math. Have a great day.

## Tuesday, October 24, 2017

### Emojis in Math

In the latest issue of The Mathematics Teacher, I found an article on using Emoji's in the math classroom. I had just had a chance to glance at the title before I had to go to work.

I had a student in my second period class who struggled with the concept of anything to the zero power is one. She kept trying to get it but just couldn't break through. So I remembered the title and started drawing different emoji's on the paper, all to the zero power.

After a few examples, she had it and could do it. This supports what the author of the article observed. If you give students the algebraic math problem in standard form, they struggle to solve it due the variables being too abstract and too far from their experiences.

When using emoji's instead, the same problem made sense to them because students relate to them since emoji's are a part of their lives. In fact, students could often solve the algebraic emoji problems in their heads, yet couldn't solve them when the same problems used variables. The use of emoji's allow students to connect with previous knowledge and begin the first steps towards algebraic understanding.

I realize the use of emoji's instead of variables is quite different but does it really matter what the representation is? I've had students suggest question marks or a square for the unknown. Yes, using letters for the variables is traditional but if students are able to gain the concepts using emoji's why not. Isn't it the concept that is important and not the representation?

This information first appeared in the blog of one of the authors. The example shown is wonderful and lots of fun. I found Solvemoji's which is an online site filled with emoji math problems that works on both computers and iPads. I had a blast working some of the problems. What is great is they have multiple levels of problems from easy to hard. If your answer is wrong, it tells you and encourages you to try again.

Step one, let them "play" at the site solving several puzzles. The second step is to begin translating the emoji math into algebraic style math with the emoji's. Third step, translating the problems into the algebraic format complete with the normal variables.

I think this is going to be a lot of fun. Let me know what you think. I love hearing from people. Have a great day.

I had a student in my second period class who struggled with the concept of anything to the zero power is one. She kept trying to get it but just couldn't break through. So I remembered the title and started drawing different emoji's on the paper, all to the zero power.

After a few examples, she had it and could do it. This supports what the author of the article observed. If you give students the algebraic math problem in standard form, they struggle to solve it due the variables being too abstract and too far from their experiences.

When using emoji's instead, the same problem made sense to them because students relate to them since emoji's are a part of their lives. In fact, students could often solve the algebraic emoji problems in their heads, yet couldn't solve them when the same problems used variables. The use of emoji's allow students to connect with previous knowledge and begin the first steps towards algebraic understanding.

I realize the use of emoji's instead of variables is quite different but does it really matter what the representation is? I've had students suggest question marks or a square for the unknown. Yes, using letters for the variables is traditional but if students are able to gain the concepts using emoji's why not. Isn't it the concept that is important and not the representation?

This information first appeared in the blog of one of the authors. The example shown is wonderful and lots of fun. I found Solvemoji's which is an online site filled with emoji math problems that works on both computers and iPads. I had a blast working some of the problems. What is great is they have multiple levels of problems from easy to hard. If your answer is wrong, it tells you and encourages you to try again.

Step one, let them "play" at the site solving several puzzles. The second step is to begin translating the emoji math into algebraic style math with the emoji's. Third step, translating the problems into the algebraic format complete with the normal variables.

I think this is going to be a lot of fun. Let me know what you think. I love hearing from people. Have a great day.

## Monday, October 23, 2017

### Google Tour Builder

I love checking out twitter on a regular basis. I learn so much and get ideas for things I can use in my math class. Late last week, there was a post on the Tour Builder from Google that is in beta form.

It allows people to create tours based on where they've been or where they want to go to. The tour incorporates google earth, photos, google street view, maps, videos to create a tour.

The reason I'm excited about this product is that it makes it easier for students to create tours based on mathematical themes. I saw a tour for Kindergarten which showed students 3 dimensional shapes in real life. There were some wonderful pieces of architecture contained within the tour. Older students could make their own versions.

Several months ago, I wrote about two tours in the UK where people walked through areas of Oxford to see mathematically based buildings, etc. Students could research mathematically inspired buildings and create their own tours via tour builder. I know that most of my students will never leave the village or at least never get farther than Anchorage. This type of activity shows them a world outside of Alaska, they will never see on their own.

The program allows people to connect slides to Google Earth, maps and sometimes Street view. It allows students to place more than one image or set of text in each slide. Students can add pictures they take or find on the internet. The program allows videos to be incorporated into the videos.

I plan to have my geometry students use their cell phones to take pictures of geometric shapes, vocabulary, etc around the village. I'll include a portion that they must take a video while explaining why they chose a certain item for their tour. Its harder out in the village because we do not have anything hire than a one story building, even at the airport with its metal buildings large enough to house front end loaders.

Think about all the possibilities:

1. Basic geometric vocabulary with pictures from local places.

2. 3 dimensional shapes from around the world.

3. Trigonometric applications such as surveying, in the real world.

4. Graphs, charts, and other real world applications.

5. Math tours of certain cities.

6. Vectors and plane trips from one place to another.

7. Taxi cab geometry.

There is this wonderful site with step by step directions to create a tour using the program and a multitude of resources geared to see what can be done with Tour Builder.

Let me know what you think. I'd love to hear. I hope to have my example put together later in the week so I can share it with you. Have a great day.

It allows people to create tours based on where they've been or where they want to go to. The tour incorporates google earth, photos, google street view, maps, videos to create a tour.

The reason I'm excited about this product is that it makes it easier for students to create tours based on mathematical themes. I saw a tour for Kindergarten which showed students 3 dimensional shapes in real life. There were some wonderful pieces of architecture contained within the tour. Older students could make their own versions.

Several months ago, I wrote about two tours in the UK where people walked through areas of Oxford to see mathematically based buildings, etc. Students could research mathematically inspired buildings and create their own tours via tour builder. I know that most of my students will never leave the village or at least never get farther than Anchorage. This type of activity shows them a world outside of Alaska, they will never see on their own.

The program allows people to connect slides to Google Earth, maps and sometimes Street view. It allows students to place more than one image or set of text in each slide. Students can add pictures they take or find on the internet. The program allows videos to be incorporated into the videos.

I plan to have my geometry students use their cell phones to take pictures of geometric shapes, vocabulary, etc around the village. I'll include a portion that they must take a video while explaining why they chose a certain item for their tour. Its harder out in the village because we do not have anything hire than a one story building, even at the airport with its metal buildings large enough to house front end loaders.

Think about all the possibilities:

1. Basic geometric vocabulary with pictures from local places.

2. 3 dimensional shapes from around the world.

3. Trigonometric applications such as surveying, in the real world.

4. Graphs, charts, and other real world applications.

5. Math tours of certain cities.

6. Vectors and plane trips from one place to another.

7. Taxi cab geometry.

There is this wonderful site with step by step directions to create a tour using the program and a multitude of resources geared to see what can be done with Tour Builder.

Let me know what you think. I'd love to hear. I hope to have my example put together later in the week so I can share it with you. Have a great day.

## Sunday, October 22, 2017

## Saturday, October 21, 2017

## Friday, October 20, 2017

### Authentic Tasks

Today, I am looking at what makes a good authentic task. Not every authentic task found on the internet are good. I know many of the performance tasks are written to provide a single answer.

Authentic tasks allow teachers a way to make math relevant for the student by asking them to make real world decisions.

When looking at authentic tasks or trying to write one, it is best to remember there is a product involved because in real life an employee usually creates something and does not complete a worksheet. The product might be something like someone in the government using census data to determine trends of growth or an environmental scientist who creates a policy paper.

When deciding if an authentic task is "real" apply the following guidelines:

1. It has a purpose and is engaging. In other words, students have to see there is real value in the task. They should want to do it.

2. Models how people solve real problems at work or in communities. They should be designed to include negotiation, planning, action, reporting, evaluating and exploring of alternatives.

3. Has students apply their knowledge by drawing on skills and strategies from different areas of mathematics.

4. Allows students to demonstrate what they know and can do. Students should be able to contribute and they should be challenged.

5. Supports multiple representations and solutions.

6. Offers opportunities for meaningful learning and allows students to use higher order thinking strategies. Students should be allowed an "aha" moment, be able to develop a number of problem solving strategies and skills, and allow for the construction and evaluation of conjectures, rules or generalizations.

7. Results in some product as a result of their deliberations. At the end, they have a tangible result.

It is easy to find a variety of authentic tasks on the internet but not all of them are realistic or meet the above criteria. Evaluate and determine if the task meets all of the criteria before assigning it. I've found worksheets having students answer questions which fall under the category of error analysis not a proper task filled with interest.

Let me know what you think. I love hearing feedback.

Authentic tasks allow teachers a way to make math relevant for the student by asking them to make real world decisions.

When looking at authentic tasks or trying to write one, it is best to remember there is a product involved because in real life an employee usually creates something and does not complete a worksheet. The product might be something like someone in the government using census data to determine trends of growth or an environmental scientist who creates a policy paper.

When deciding if an authentic task is "real" apply the following guidelines:

1. It has a purpose and is engaging. In other words, students have to see there is real value in the task. They should want to do it.

2. Models how people solve real problems at work or in communities. They should be designed to include negotiation, planning, action, reporting, evaluating and exploring of alternatives.

3. Has students apply their knowledge by drawing on skills and strategies from different areas of mathematics.

4. Allows students to demonstrate what they know and can do. Students should be able to contribute and they should be challenged.

5. Supports multiple representations and solutions.

6. Offers opportunities for meaningful learning and allows students to use higher order thinking strategies. Students should be allowed an "aha" moment, be able to develop a number of problem solving strategies and skills, and allow for the construction and evaluation of conjectures, rules or generalizations.

7. Results in some product as a result of their deliberations. At the end, they have a tangible result.

It is easy to find a variety of authentic tasks on the internet but not all of them are realistic or meet the above criteria. Evaluate and determine if the task meets all of the criteria before assigning it. I've found worksheets having students answer questions which fall under the category of error analysis not a proper task filled with interest.

Let me know what you think. I love hearing feedback.

## Thursday, October 19, 2017

### Word Problems in Context.

Unfortunately, too many word problems found in textbooks tend to be artificial and constructed to meet the math being taught at that moment. The unfortunate side effect to this is students look for the math to apply rather than starting with the word problem itself.

Lets look at the process involved in solving real world problems starting with the context rather than the math.

We are given a mathematical problem in context. From that we formulate the mathematical application and its uses to the real world problem. This includes taking the situation and translating it into a form which we can identify the mathematics needed to solve it. The next step is to use any mathematical formulas, processes, procedures, reasoning, facts or tools required to find the solution. The final step is to interpret and or evaluate the answer to determine if the solution is reasonable.

This is directly opposite of the normal way of doing things where we teach some math, have them practice the math, finally applying the math in a situation where the math has already been set up for the student.

We as math teachers can step away from the textbooks, to provide open ended real texts and real situations so students have a chance to make connections between math and its use in the real world. We can provide students with those messy problems and help them learn to extract the information needed to identify the mathematics needed to solve it, and when students stumble, use that to determine what needs to be taught.

As part of providing these types of experiences, there has been a creation of authentic tasks which require students to demonstrate their knowledge by applying it to real world problems. A good task bridges the gap between the classroom and understanding its applications outside of the classroom.

Tomorrow, I'll be discussing how a good authentic or real world task is built. Let me know what you think.

Lets look at the process involved in solving real world problems starting with the context rather than the math.

We are given a mathematical problem in context. From that we formulate the mathematical application and its uses to the real world problem. This includes taking the situation and translating it into a form which we can identify the mathematics needed to solve it. The next step is to use any mathematical formulas, processes, procedures, reasoning, facts or tools required to find the solution. The final step is to interpret and or evaluate the answer to determine if the solution is reasonable.

This is directly opposite of the normal way of doing things where we teach some math, have them practice the math, finally applying the math in a situation where the math has already been set up for the student.

We as math teachers can step away from the textbooks, to provide open ended real texts and real situations so students have a chance to make connections between math and its use in the real world. We can provide students with those messy problems and help them learn to extract the information needed to identify the mathematics needed to solve it, and when students stumble, use that to determine what needs to be taught.

As part of providing these types of experiences, there has been a creation of authentic tasks which require students to demonstrate their knowledge by applying it to real world problems. A good task bridges the gap between the classroom and understanding its applications outside of the classroom.

Tomorrow, I'll be discussing how a good authentic or real world task is built. Let me know what you think.

## Wednesday, October 18, 2017

### Geogebra AR (Augmented Reality)

The other day, I heard that Geogebra now offers an augmented reality app which superimposes shapes over the background seen by the camera. You snap and can capture the scene.

When you point your camera at something you can choose to insert basic solids, the Penrose triangle, the Sierpinski pyramid, a football, 3d function, Klein's bottle, and a ruled surface.

This first shot is one of the shapes from the basic solids. Before I took the picture, there is a set of instructions telling students to take pictures of pyramids, prisms or consist of only triangles.

In addition, with two fingers, I can move the group of shapes around so I can see all sides of each element in the photo.

I can move in closer as you can see with the photo to the right. I can get close enough to see everything in detail.

I can move out to see the whole shape in context so there is a lot of flexibility. By moving in closer, I can take a picture of the one or move out and get a picture of the whole group.

The shot to the left shows the group looking sideways so you see it as if they are all in the same plane. Side note here: I took all these photos from my front porch, overlooking the lake.

The shapes float over the grassy edge between the marshy area and the lake itself. Since you can move the shapes around, it is possible to take photos of each shape from all sides.

The photo to the right was taken looking straight down at my carpet. This indicates the app is able to orient the objects to the direction the camera is facing.

From this way, you can see what the objects look from overhead which makes it easier for students to see how pieces fit together.

I usually have students create their own nets for various shapes but they often have trouble because they have never really paid attention to how the shapes are put together.

This app would allow them to take one shape, explore it before working on creating the netting needed to recreate the shape out of paper.

Unfortunately, I have not found any developed lesson plans other than the instructions found at the bottom of view screen.

Keep your eyes peeled because next week, I'll be doing a column on using Augmented Reality in the math classroom.

I hope every one has a great day. Let me know what you think.

When you point your camera at something you can choose to insert basic solids, the Penrose triangle, the Sierpinski pyramid, a football, 3d function, Klein's bottle, and a ruled surface.

This first shot is one of the shapes from the basic solids. Before I took the picture, there is a set of instructions telling students to take pictures of pyramids, prisms or consist of only triangles.

In addition, with two fingers, I can move the group of shapes around so I can see all sides of each element in the photo.

I can move in closer as you can see with the photo to the right. I can get close enough to see everything in detail.

I can move out to see the whole shape in context so there is a lot of flexibility. By moving in closer, I can take a picture of the one or move out and get a picture of the whole group.

The shot to the left shows the group looking sideways so you see it as if they are all in the same plane. Side note here: I took all these photos from my front porch, overlooking the lake.

The shapes float over the grassy edge between the marshy area and the lake itself. Since you can move the shapes around, it is possible to take photos of each shape from all sides.

The photo to the right was taken looking straight down at my carpet. This indicates the app is able to orient the objects to the direction the camera is facing.

From this way, you can see what the objects look from overhead which makes it easier for students to see how pieces fit together.

I usually have students create their own nets for various shapes but they often have trouble because they have never really paid attention to how the shapes are put together.

This app would allow them to take one shape, explore it before working on creating the netting needed to recreate the shape out of paper.

Unfortunately, I have not found any developed lesson plans other than the instructions found at the bottom of view screen.

Keep your eyes peeled because next week, I'll be doing a column on using Augmented Reality in the math classroom.

I hope every one has a great day. Let me know what you think.

## Tuesday, October 17, 2017

### Mathematical Activities Using the Aurora.

The one thing I love about living in Alaska is seeing the Auroral displays in winter. Yes, it can be extremely cold but many times the displays are so awesome, its worth standing in -40 degree weather to enjoy.

Its always nice to have activities lined up to help students understand more about the mathematics behind the aurora.

Before starting either lesson, I'd show the videos from this page as a way of introducing these beautiful phenomena to students who might not have a chance to see the lights in person.

For instance, NASA has a lovely 56 page activity guide geared for grades 7 and 8 with several mathematical activities such as plotting satellite data on a polar map to see where the auroral belt is located. Another uses geometry to find observing latitude for auroral displays. There is an activity using a clinometer to find the height of an object in the classroom while another has students using triangulation to find the height of the aurora. There are four more activities which use math and deal with the aurora.

Each activity comes complete with objectives, sample questions, a list of materials, worksheets, demonstration information and teacher notes. Although it is geared for 7th and 8th graders, this could easily be used in the high school with little alternation.

The Utah Education Network has a great activity that uses a circular grid to plot zones of auroral activities. It takes the students step by step through the activity to analyze the northern nights. When they've completed this one, there is a worksheet for students to use to analyze the southern lights in the same way. I like that this activity has them using a geographic circular grid to provide a reference.

Most of the activities in both sets of activities have students learning to find points on a geographic coordinate grid which gives them exposure to a different use of coordinate points. In this case, the x and y values represent longitude and latitude, a use most of our students are never exposed to . It puts the graphing into a real contextual use.

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

Its always nice to have activities lined up to help students understand more about the mathematics behind the aurora.

Before starting either lesson, I'd show the videos from this page as a way of introducing these beautiful phenomena to students who might not have a chance to see the lights in person.

For instance, NASA has a lovely 56 page activity guide geared for grades 7 and 8 with several mathematical activities such as plotting satellite data on a polar map to see where the auroral belt is located. Another uses geometry to find observing latitude for auroral displays. There is an activity using a clinometer to find the height of an object in the classroom while another has students using triangulation to find the height of the aurora. There are four more activities which use math and deal with the aurora.

Each activity comes complete with objectives, sample questions, a list of materials, worksheets, demonstration information and teacher notes. Although it is geared for 7th and 8th graders, this could easily be used in the high school with little alternation.

The Utah Education Network has a great activity that uses a circular grid to plot zones of auroral activities. It takes the students step by step through the activity to analyze the northern nights. When they've completed this one, there is a worksheet for students to use to analyze the southern lights in the same way. I like that this activity has them using a geographic circular grid to provide a reference.

Most of the activities in both sets of activities have students learning to find points on a geographic coordinate grid which gives them exposure to a different use of coordinate points. In this case, the x and y values represent longitude and latitude, a use most of our students are never exposed to . It puts the graphing into a real contextual use.

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

## Monday, October 16, 2017

### The Droste Effect.

Over the weekend, I found a cool app for my ipad called Hyperdroste. Droste is a technique which causes smaller images to appear within the original one in a recursive pattern.

The name originated in Holland. Think of a picture repeated within itself. It was named after the chocolate of the same name who first used the effect on a package of its cocoa. The picture showed a nurse holding a tray with a box of Droste's cocoa and cup of cocoa which showed the nurse with a tray holding a box of Droste's cocoa and cup of cocoa, and one and on.

They were the first to use it but if you look around you'll find the effect on record covers, food advertisements, camera ads, etc.

The picture to the left is an example of the effect. The effect is also seen in some of Escher's artwork.

The mathematics for this technique was finally worked out in 2003. Several mathematicians at Leiden University used Escher's Print Gallery drawing to figure out the mathematics involved in this technique because Escher used it a lot. The thing about the Print Gallery creation is that the center has a white space asking people to finish the pattern so these mathematicians did.

Escher apparently took square grids and transposed them onto a curved grid. They discovered the basic technique is first stage uses a transformation of z to log(z) for base e logarithm, the second uses rotation and scaling or rotation dilation, while the third stage uses exponentiation of z to e^z to create the spiral effects. After more checking, they concluded Escher used a scale factor of 18 and a 160 degree rotation for the second step.

It turns out he stumbled across a mathematical concept called Conformal Mapping which is an angle preserving transformation. This concept allows people to transform anything and still get a recognizable image.

Unfortunately, I cannot find any instructions for doing this by hand. I did find a wonderful description of the mathematics used in this process at this site. There are numerous sites with tutorials for using Photoshop, Adobe, and other programs but there are a few sites online that provide free software for students to do this. One is PhotoSpiralysis which works on iPads and allows students to play with images.

I played with this program on my Mac and here are two photos I created using the program.

The first photo above is plain with the effect applied so you see pumpkins within pumpkins, within pumpkins.

Check around and you'll find other apps, programs, etc which you can have students use to create these. As always, let me know what you think.

The name originated in Holland. Think of a picture repeated within itself. It was named after the chocolate of the same name who first used the effect on a package of its cocoa. The picture showed a nurse holding a tray with a box of Droste's cocoa and cup of cocoa which showed the nurse with a tray holding a box of Droste's cocoa and cup of cocoa, and one and on.

They were the first to use it but if you look around you'll find the effect on record covers, food advertisements, camera ads, etc.

The picture to the left is an example of the effect. The effect is also seen in some of Escher's artwork.

The mathematics for this technique was finally worked out in 2003. Several mathematicians at Leiden University used Escher's Print Gallery drawing to figure out the mathematics involved in this technique because Escher used it a lot. The thing about the Print Gallery creation is that the center has a white space asking people to finish the pattern so these mathematicians did.

Escher apparently took square grids and transposed them onto a curved grid. They discovered the basic technique is first stage uses a transformation of z to log(z) for base e logarithm, the second uses rotation and scaling or rotation dilation, while the third stage uses exponentiation of z to e^z to create the spiral effects. After more checking, they concluded Escher used a scale factor of 18 and a 160 degree rotation for the second step.

It turns out he stumbled across a mathematical concept called Conformal Mapping which is an angle preserving transformation. This concept allows people to transform anything and still get a recognizable image.

Unfortunately, I cannot find any instructions for doing this by hand. I did find a wonderful description of the mathematics used in this process at this site. There are numerous sites with tutorials for using Photoshop, Adobe, and other programs but there are a few sites online that provide free software for students to do this. One is PhotoSpiralysis which works on iPads and allows students to play with images.

I played with this program on my Mac and here are two photos I created using the program.

The first photo above is plain with the effect applied so you see pumpkins within pumpkins, within pumpkins.

The second photo, above, has taken the same photo and spiraled it so it looks more like a nebula.

Check around and you'll find other apps, programs, etc which you can have students use to create these. As always, let me know what you think.

## Sunday, October 15, 2017

## Saturday, October 14, 2017

## Friday, October 13, 2017

### Sweets to Teach Statistics.

In the past, I've used M&M's or Skittles to teach students about pie charts. They've sorted the candy into each color before calculating the percent of each color in the package. Once they had the information in fraction, decimal, and percent forms, they were ready to create pie charts, bar graphs and other methods of visualizing the information.

I am about 2 years behind in reading my NCTM magazines. In the September 2016 issue of Mathematics teacher, a title involving tootsie pops and statistics caught my attention.

A quick search shows a variety of candy based activities designed to teach both probability or statistics. The candy element creates interest especially if they know they will be able to eat the sweets when the lesson is completed. I will share a few of the resources and links so you can explore the activities in more detail.

The Society For The Teaching of Psychology has a wonderful pdf filled with 9 activities and demonstrations designed to teach basic statistics and research methods. These activities cover everything from population and samples to components of experiments, to central tendencies and probability. Only one of the actual activities uses candies, number eight, but the others use music, dating, football, etc.

The eighth activity uses skittles to help teach more about population, sampling, probabilities and includes vocabulary. There are 5 questions asking them to calculate the probability for finding skittles that are not red, selecting three oranges from the bag and other situations.

How about having two bags of small bars of candy. In the first bag you tell students there is a equal mix of Mars, Twix, and Snickers. You tell students that if they can guess the candy you just drew out they get it. What they don't know is that you placed only Mars bars in the bag. After about 6 bars, see if the students have noticed things are not quite right. This leads to a wonderful discussion of the probability of getting three in a row of the same candy if it has equal amounts of the three candy bars.

Then have them guess the candy you'll be pulling out of the second bag. This bag is filled with an equal mix of Twix and Snickers with no Mars bars. By now students are suspicious. This activity gives you a chance to discuss null hypothesis, hypothesis testing, and inference. The above activity came from the Learn and Teach Statistics blog.

From another Teach Psychology website, we have access to a power point presentation containing all the information to run the activity on Samples Representing The Population using M & M's. This activity begins by having all the students take their bag of M & M's, sort them by color, and calculate the percentage each color represented in the bag.

The students go around visiting with other students to write down their results and using all 6 samples to calculate the percent of each color over the larger sample. Students are asked to compare the results of the small samples with the larger population and then explain if the percentages changed. At the end, they compare their results with the official M & M percentages. The point of this exercise is to show students the larger the population, the more accurate the results.

I'll cover the Tootsie Pop exercise next week. Let me know what you think of these.

Have a great weekend.

I am about 2 years behind in reading my NCTM magazines. In the September 2016 issue of Mathematics teacher, a title involving tootsie pops and statistics caught my attention.

A quick search shows a variety of candy based activities designed to teach both probability or statistics. The candy element creates interest especially if they know they will be able to eat the sweets when the lesson is completed. I will share a few of the resources and links so you can explore the activities in more detail.

The Society For The Teaching of Psychology has a wonderful pdf filled with 9 activities and demonstrations designed to teach basic statistics and research methods. These activities cover everything from population and samples to components of experiments, to central tendencies and probability. Only one of the actual activities uses candies, number eight, but the others use music, dating, football, etc.

The eighth activity uses skittles to help teach more about population, sampling, probabilities and includes vocabulary. There are 5 questions asking them to calculate the probability for finding skittles that are not red, selecting three oranges from the bag and other situations.

How about having two bags of small bars of candy. In the first bag you tell students there is a equal mix of Mars, Twix, and Snickers. You tell students that if they can guess the candy you just drew out they get it. What they don't know is that you placed only Mars bars in the bag. After about 6 bars, see if the students have noticed things are not quite right. This leads to a wonderful discussion of the probability of getting three in a row of the same candy if it has equal amounts of the three candy bars.

Then have them guess the candy you'll be pulling out of the second bag. This bag is filled with an equal mix of Twix and Snickers with no Mars bars. By now students are suspicious. This activity gives you a chance to discuss null hypothesis, hypothesis testing, and inference. The above activity came from the Learn and Teach Statistics blog.

From another Teach Psychology website, we have access to a power point presentation containing all the information to run the activity on Samples Representing The Population using M & M's. This activity begins by having all the students take their bag of M & M's, sort them by color, and calculate the percentage each color represented in the bag.

The students go around visiting with other students to write down their results and using all 6 samples to calculate the percent of each color over the larger sample. Students are asked to compare the results of the small samples with the larger population and then explain if the percentages changed. At the end, they compare their results with the official M & M percentages. The point of this exercise is to show students the larger the population, the more accurate the results.

I'll cover the Tootsie Pop exercise next week. Let me know what you think of these.

Have a great weekend.

## Thursday, October 12, 2017

### Math Magic

Have you ever thought about using the magic of mathematics to help students understand math better and to increase interest? I hadn't until I read a couple of articles on it.

The mathematical tricks in math can be divided into three categories.

The first type are the ones which place an operation or two on an original number and end up with a given number. An example is:

Think of a number between 1 and 100.

Multiply your number by 4

Add 12

Multiply this number by 2

Add 16

Divide the number by 8

Subtract your original number.

Your answer is 5.

The second type are those which recombine and rearrange digits. An example is:

Write a three digit number using three different digits.

Mix up the digits so you get a different three digit number.

Subtract the smaller number from the larger number.

Add the digits in the difference to get a one digit number.

Subtract 5 to get the final number. The answer should be 4.

The third type is where the beginning number is the final number. An example of this one is:

Write down the year of your birth

Double it.

Add 5

Multiply by 50

Add your age

Add 365

Subtract 615

Your answer should have 6 digits. The first four are the year of your birth while the last two are your age. I tried it and it worked beautifully.

The great thing about these tricks and others you can find on the internet and in books, is the fact they can be translated into algebraic equations for each step. This provides a direct link between algebra and the way these tricks work.

Math magic can helps students learn more about expressions, variables, equal signs, functions, and seeing different types of equations are solved using different algorithms. If you select the correct magic tricks you can use them to introduce function notation and inverse functions.

In case you wondered, yes, I tried it in all my math classes yesterday and it was absolutely successful. If I had the stuff, I would have gone in with my magic hat, magic wand, and put on a real show. The students were amazed at the results. They laughed and payed attention. They were fully involved. They want to do it again sometime.

I got all these magic tricks from this issue of Education World.

Let me know what you think. I'd love to hear. Have a good day.

The mathematical tricks in math can be divided into three categories.

The first type are the ones which place an operation or two on an original number and end up with a given number. An example is:

Think of a number between 1 and 100.

Multiply your number by 4

Add 12

Multiply this number by 2

Add 16

Divide the number by 8

Subtract your original number.

Your answer is 5.

The second type are those which recombine and rearrange digits. An example is:

Write a three digit number using three different digits.

Mix up the digits so you get a different three digit number.

Subtract the smaller number from the larger number.

Add the digits in the difference to get a one digit number.

Subtract 5 to get the final number. The answer should be 4.

The third type is where the beginning number is the final number. An example of this one is:

Write down the year of your birth

Double it.

Add 5

Multiply by 50

Add your age

Add 365

Subtract 615

Your answer should have 6 digits. The first four are the year of your birth while the last two are your age. I tried it and it worked beautifully.

The great thing about these tricks and others you can find on the internet and in books, is the fact they can be translated into algebraic equations for each step. This provides a direct link between algebra and the way these tricks work.

Math magic can helps students learn more about expressions, variables, equal signs, functions, and seeing different types of equations are solved using different algorithms. If you select the correct magic tricks you can use them to introduce function notation and inverse functions.

In case you wondered, yes, I tried it in all my math classes yesterday and it was absolutely successful. If I had the stuff, I would have gone in with my magic hat, magic wand, and put on a real show. The students were amazed at the results. They laughed and payed attention. They were fully involved. They want to do it again sometime.

I got all these magic tricks from this issue of Education World.

Let me know what you think. I'd love to hear. Have a good day.

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