What is Scientific Notation?
Scientific notation is a way to represent a number (large or small) that allows us to give a "shorthand way" of writing out a number. Imagine having to write out the number 938,000,000,000, ten times. It would not take you long before you realized that writing it this way is a lot of work. Scientific notation would allow us to represent the number 938,000,000,000 by saying it is equivalent to 9.38 X 10^11. This has obvious advantages when working with numbers all day long.
This method can be used with incredibly small numbers as well (the difference being the exponent in the notation). Instead of having a positive exponent, you would have a negative.
So how do we do it?
It's really quite simple. When representing a large number, move your decimal place over to the right of the first non-zero digit (in this case, 9). Counting the number of places you move the decimal, this will give you the exponent you will need to use. We used the exponent (11) because we moved the decimal 11 places to the left.
Let's go the opposite way:
Let the new number be the following: 0.00000682. Represent it using scientific notation.
6.82 X 10^-6. Since 6 is the first non-zero digit, we place the decimal to the right of that place value. This allows us to move the decimal 6 places to the right - meaning the exponent is negative.
The following is a YouTube video explaining scientific notation:
Scientific Notation
Monday, July 22, 2013
Thinking Blocks
What are they?
Thinking blocks is a way to represent mathematical problems by using building blocks to represent the elements of the problem. Using thinking blocks is a great tool. I really like how this website (Thinking Blocks) lays out a description should you get an answer incorrect. It makes it so that you can easily see your mistake, and allows you to correct that mistake. I challenge you to try out some of the sections of the website and play with the problem builders.
After completing the sections, I thought that this would be a very useful tool in helping students understand the concepts of addition, subtraction, etc., and you could even incorporate algebra into it by representing the unknown variable with "x" instead of with a question mark. This seems to be more important at increasingly earlier ages in schoolchildren. Recently, I know a local school district began classifying what was 6th grade math level to now be what should be taught in the 5th grade levels.
All of these stages incorporate addition and subtraction skills, but the more advanced stages (Change 2/Compare 3) I believe would be better suited for 4th/5th grade application. The earlier stages I could see being used in earlier elementary years (2nd/3rd grade). You really would have to gauge this type of lesson off of your curriculum and what is to be covered by state standards. I agree with Jillian, that you could use this tool for a variety of applications - fractions, multiplication, etc. Overall, a great tool to use in the classroom!
Try it out, see what you think, and most importantly, have fun!
Thinking blocks is a way to represent mathematical problems by using building blocks to represent the elements of the problem. Using thinking blocks is a great tool. I really like how this website (Thinking Blocks) lays out a description should you get an answer incorrect. It makes it so that you can easily see your mistake, and allows you to correct that mistake. I challenge you to try out some of the sections of the website and play with the problem builders.
After completing the sections, I thought that this would be a very useful tool in helping students understand the concepts of addition, subtraction, etc., and you could even incorporate algebra into it by representing the unknown variable with "x" instead of with a question mark. This seems to be more important at increasingly earlier ages in schoolchildren. Recently, I know a local school district began classifying what was 6th grade math level to now be what should be taught in the 5th grade levels.
All of these stages incorporate addition and subtraction skills, but the more advanced stages (Change 2/Compare 3) I believe would be better suited for 4th/5th grade application. The earlier stages I could see being used in earlier elementary years (2nd/3rd grade). You really would have to gauge this type of lesson off of your curriculum and what is to be covered by state standards. I agree with Jillian, that you could use this tool for a variety of applications - fractions, multiplication, etc. Overall, a great tool to use in the classroom!
Try it out, see what you think, and most importantly, have fun!
Technologically Dependent
The Concept
In today's hustle and bustle world, with seemingly everyone multitasking, over scheduled, and burning the candles at both ends, have we become too dependent on technology simply because it is easier, or is it because we cannot devote the time to working problems out by hand?
I believe that it is a little of both, but that we also should not allow ourselves to fall away from teaching, learning, and understanding how to work concepts by hand. What are we to do if technology fails to support what we demand of it? Countless times I have been writing a paper or working on problems while doing my online homework and I have been thankful that I have backed up my work on my computer - as it seems that when you need the work most, there is a technological hiccup.
How do we stay ahead of technology?
The best way for us to stay ahead of technology it to continue teaching children the concepts, but blending it in a way that they can be done with or without technology. It seems that most who try to answer this question say it is "easier" to go away from the "old school" curriculum in general (such as in a previous blog I had wrote about long division). Do I agree that technology has streamlined and simplified many things in our lives? Absolutely. I'm an avid techie just like most who have grown up in my generation. I do feel that there is a strong reason to continue teaching the concepts by hand: Brains are biological and permanent. Computers can be unplugged, deleted, and disassembled.
For those of us who are studying to be educators: What is to become of us if we are simply all using computers (solely) for our lesson plans? Are we all going to be referred to as a type of curriculum "programmer"? Maybe an Electrical Educator? I think that some of the everyday concepts we know and love are taught through live, personal interaction with one another.
The following are a couple of interesting links to teaching with or without the use of technology:
Can You Teach Without Technology?
Evolution or Revolution?
In today's hustle and bustle world, with seemingly everyone multitasking, over scheduled, and burning the candles at both ends, have we become too dependent on technology simply because it is easier, or is it because we cannot devote the time to working problems out by hand?
I believe that it is a little of both, but that we also should not allow ourselves to fall away from teaching, learning, and understanding how to work concepts by hand. What are we to do if technology fails to support what we demand of it? Countless times I have been writing a paper or working on problems while doing my online homework and I have been thankful that I have backed up my work on my computer - as it seems that when you need the work most, there is a technological hiccup.
How do we stay ahead of technology?
The best way for us to stay ahead of technology it to continue teaching children the concepts, but blending it in a way that they can be done with or without technology. It seems that most who try to answer this question say it is "easier" to go away from the "old school" curriculum in general (such as in a previous blog I had wrote about long division). Do I agree that technology has streamlined and simplified many things in our lives? Absolutely. I'm an avid techie just like most who have grown up in my generation. I do feel that there is a strong reason to continue teaching the concepts by hand: Brains are biological and permanent. Computers can be unplugged, deleted, and disassembled.
For those of us who are studying to be educators: What is to become of us if we are simply all using computers (solely) for our lesson plans? Are we all going to be referred to as a type of curriculum "programmer"? Maybe an Electrical Educator? I think that some of the everyday concepts we know and love are taught through live, personal interaction with one another.
The following are a couple of interesting links to teaching with or without the use of technology:
Can You Teach Without Technology?
Evolution or Revolution?
Sinking a ship: Summer storms on Lake Osakis - Triangular Travel.
Good afternoon!
After a seemingly nonexistent spring, it appeared that summer was here - if only to arrive in mid June. When summer came to Minnesota, this year it was all or nothing. From frozen lakes on fishing opener to some of the most violent storms the state has seen across the northwest plains. Lake Osakis, was unfortunately about to discover just how violent summer can be.
The Story:
It was six in the morning, and I had been tossing and turning for the better part of a couple hours (probably dreading the homework I had in store for myself). All of a sudden my phone rings and my good friend and former teacher, Ron, is frantic. "My neighbor at the lake called me and said my boat is totaled! We have to get up there right now!" I agreed and informed him that we needed to be back by one in the afternoon, as I had an appointment that I couldn't break. He arrived at my home, and we jumped in the truck and hurried to his cabin.
Upon our arrival, we couldn't believe our eyes. Nearly the entire South shore of the lake looked like it had been hit by a hurricane - come to find out that the winds were 90+ miles per hour, with waves reaching 5-6 feet. The entire South shore of boats, docks, and boat lifts looked like the backyard of an aluminum scrapyard. The task at hand: Get the boat off of the bottom of the lake, out from under a collapsed dock, and onto the trailer. The difficult part: the boat was nearly split in half and was full of seaweed, water, mud, and who knows what else. The only way we had to get the boat out was to hoist it by crane (which was already one scene). The boat, at 17.5 feet long and 7.5 feet wide had to balance on two straps that were approximately 6" wide and 1/4" thick while being hoisted several feet up into the air and over the embankment
The Mathematical Concept:
The embankment was 12 feet below the road level, and 14 feet from the road out to where the boat was sitting. How far was the boat moved from the point it was sitting, to the point where it was placed on the trailer? (The trailer was approximately 2 feet higher than the surface of the road, and 2 feet in). So: 14 feet above road level, and 16 feet in from the boats position. What is the hypotenuse length (the length of travel in the air)?
Use the Pythagorean theorem: a^2 + b^2 = c^2
14^2 + 16^2
196 + 256 = c^2
c^2 = 452. The square root of 452 is approximately 21.3 feet.
Why is this interesting?
This concept is interesting to me because I found it a challenge to figure out the distance the boat traveled. I pictured the boat sliding up planks on the embankment (which we did not use). And for me to understand how to figure out the distance it traveled was similar to me having to figure the distance (in length) we would have needed had we used planks. This type of math is practical in every day application, and it shows you that you never know when you may need it.
Enjoy a couple of pictures from the damage!
After a seemingly nonexistent spring, it appeared that summer was here - if only to arrive in mid June. When summer came to Minnesota, this year it was all or nothing. From frozen lakes on fishing opener to some of the most violent storms the state has seen across the northwest plains. Lake Osakis, was unfortunately about to discover just how violent summer can be.
The Story:
It was six in the morning, and I had been tossing and turning for the better part of a couple hours (probably dreading the homework I had in store for myself). All of a sudden my phone rings and my good friend and former teacher, Ron, is frantic. "My neighbor at the lake called me and said my boat is totaled! We have to get up there right now!" I agreed and informed him that we needed to be back by one in the afternoon, as I had an appointment that I couldn't break. He arrived at my home, and we jumped in the truck and hurried to his cabin.
Upon our arrival, we couldn't believe our eyes. Nearly the entire South shore of the lake looked like it had been hit by a hurricane - come to find out that the winds were 90+ miles per hour, with waves reaching 5-6 feet. The entire South shore of boats, docks, and boat lifts looked like the backyard of an aluminum scrapyard. The task at hand: Get the boat off of the bottom of the lake, out from under a collapsed dock, and onto the trailer. The difficult part: the boat was nearly split in half and was full of seaweed, water, mud, and who knows what else. The only way we had to get the boat out was to hoist it by crane (which was already one scene). The boat, at 17.5 feet long and 7.5 feet wide had to balance on two straps that were approximately 6" wide and 1/4" thick while being hoisted several feet up into the air and over the embankment
The Mathematical Concept:
The embankment was 12 feet below the road level, and 14 feet from the road out to where the boat was sitting. How far was the boat moved from the point it was sitting, to the point where it was placed on the trailer? (The trailer was approximately 2 feet higher than the surface of the road, and 2 feet in). So: 14 feet above road level, and 16 feet in from the boats position. What is the hypotenuse length (the length of travel in the air)?
Use the Pythagorean theorem: a^2 + b^2 = c^2
14^2 + 16^2
196 + 256 = c^2
c^2 = 452. The square root of 452 is approximately 21.3 feet.
Why is this interesting?
This concept is interesting to me because I found it a challenge to figure out the distance the boat traveled. I pictured the boat sliding up planks on the embankment (which we did not use). And for me to understand how to figure out the distance it traveled was similar to me having to figure the distance (in length) we would have needed had we used planks. This type of math is practical in every day application, and it shows you that you never know when you may need it.
Enjoy a couple of pictures from the damage!
Long Division: So Long Ago!
Hello all!
We all have used division in our daily lives - no matter how we look at it we use it. Whether it is dividing a dollar into equal increments, an apple into individual slices, or our children into two separate halves of the car. Something that we all have seemingly forgotten how to do is long division. I know I always had trouble in grade school with long division, and after talking to several friends of mine, a lot of them also struggle, nor do they remember how to actually perform the operation. Why would something like this be important to know?
Is it practical?
Long division (although it may seem that it is easier to take out our smart phone, open our calculator app, and punch the numbers in) is still a practical knowledge to have. What if your smart phone has a dead battery? Are you going to walk around aimlessly searching for a calculator or a phone charger? By the time you find one, you may have figured out your division problem, and done a couple of more just for fun and practice! I think that long division is more practical than some of the things we take for granted. We can use it in everything from dividing money for our finances and investments, to engineering problems. For those of us who are studying to manage money or design processes (mechanical, chemical, or electrical), this can prove to be a great tool!
A little review:
The following is a link to a YouTube Video that explains the process of long division. There are several steps to doing the problems, but in reality, the steps are really quite simple. It has taken me nearly 20 years to understand the concept and its' importance, but now I definitely see it!
Long Division Made Easy
Curious about the future of long division?
The following is a link to the website Wikipedia that I found interesting information on. One of the most important things I think we should focus on is the section stating that long division may be a thing of the past. Is this a skill we want to continue teaching to our kids, or are we too technologically dependent to consider it anymore?
Long Division: Wikipedia
We all have used division in our daily lives - no matter how we look at it we use it. Whether it is dividing a dollar into equal increments, an apple into individual slices, or our children into two separate halves of the car. Something that we all have seemingly forgotten how to do is long division. I know I always had trouble in grade school with long division, and after talking to several friends of mine, a lot of them also struggle, nor do they remember how to actually perform the operation. Why would something like this be important to know?
Is it practical?
Long division (although it may seem that it is easier to take out our smart phone, open our calculator app, and punch the numbers in) is still a practical knowledge to have. What if your smart phone has a dead battery? Are you going to walk around aimlessly searching for a calculator or a phone charger? By the time you find one, you may have figured out your division problem, and done a couple of more just for fun and practice! I think that long division is more practical than some of the things we take for granted. We can use it in everything from dividing money for our finances and investments, to engineering problems. For those of us who are studying to manage money or design processes (mechanical, chemical, or electrical), this can prove to be a great tool!
A little review:
The following is a link to a YouTube Video that explains the process of long division. There are several steps to doing the problems, but in reality, the steps are really quite simple. It has taken me nearly 20 years to understand the concept and its' importance, but now I definitely see it!
Long Division Made Easy
Curious about the future of long division?
The following is a link to the website Wikipedia that I found interesting information on. One of the most important things I think we should focus on is the section stating that long division may be a thing of the past. Is this a skill we want to continue teaching to our kids, or are we too technologically dependent to consider it anymore?
Long Division: Wikipedia
Sunday, June 30, 2013
Estimation or Geusstimation?
In today's world, we are constantly bombarded with numbers - whether it is on the price tag of a t-shirt we would like to buy, a bill that we are sending off in the mail, or on a menu while ordering our favorite lunch. How we implement estimation in our daily lives is mostly autonomic - we do so without even putting much effort into it. Is it imperative that we continue teaching these skills to our youth in school? I would be inclined to say that it is indeed important.
On a recent cross-country motorcycle trip that I had taken, I found myself looking at the road signs in comparison to my odometer, guessing how many more miles I had to go between towns (in order to ensure that I could fill my gas tank). I spent much of the trip doing automatic math in my head - rounding numbers, adding, subtracting, and even dividing when I filled my tank to approximate the miles per gallon I was getting. All of this is thanks to the practice I received in school learning rounding and estimation techniques. Take distance from destination "A", to destination "B" as being 106 miles. Then from destination "B" to destination "C" as 84 miles. Now let's say that I am able to go 175 miles per tank of fuel. Will I be able to make it from destination A to C? If we use the rounding method, Distance AB is approximately 110 miles, and distance BC is approximately 80. So, AB+BC = 110+80 = 190. 190 total miles - 175 miles per tank = 15 miles difference, so the answer then is no. I would be better off stopping for fuel at destination B, to be safe.
Without the basic mathematical skills to add, subtract, round, and estimate, our daily lives would seem to be much more complicating. Imagine having to memorize complex formulas, or having to stop to get out your calculator every time you needed to figure out if you needed gas (not just by going on the gauge on your dashboard of your car). These skills are essential for us to learn as we continue through not only our educational careers, but throughout our lives.
The following is a video I found in a YouTube search that should help should you have any questions on the estimation skills that are being taught in many of today's elementary schools.
http://www.youtube.com/watch?v=rbw5ptdeRsw
The following link is to an article released by the NACD (National Association for Child Development) that discusses many reasons and benefits to teaching our children estimation techniques
http://nacd.org/newsletter/0309_estimation.php
On a recent cross-country motorcycle trip that I had taken, I found myself looking at the road signs in comparison to my odometer, guessing how many more miles I had to go between towns (in order to ensure that I could fill my gas tank). I spent much of the trip doing automatic math in my head - rounding numbers, adding, subtracting, and even dividing when I filled my tank to approximate the miles per gallon I was getting. All of this is thanks to the practice I received in school learning rounding and estimation techniques. Take distance from destination "A", to destination "B" as being 106 miles. Then from destination "B" to destination "C" as 84 miles. Now let's say that I am able to go 175 miles per tank of fuel. Will I be able to make it from destination A to C? If we use the rounding method, Distance AB is approximately 110 miles, and distance BC is approximately 80. So, AB+BC = 110+80 = 190. 190 total miles - 175 miles per tank = 15 miles difference, so the answer then is no. I would be better off stopping for fuel at destination B, to be safe.
Without the basic mathematical skills to add, subtract, round, and estimate, our daily lives would seem to be much more complicating. Imagine having to memorize complex formulas, or having to stop to get out your calculator every time you needed to figure out if you needed gas (not just by going on the gauge on your dashboard of your car). These skills are essential for us to learn as we continue through not only our educational careers, but throughout our lives.
The following is a video I found in a YouTube search that should help should you have any questions on the estimation skills that are being taught in many of today's elementary schools.
http://www.youtube.com/watch?v=rbw5ptdeRsw
The following link is to an article released by the NACD (National Association for Child Development) that discusses many reasons and benefits to teaching our children estimation techniques
http://nacd.org/newsletter/0309_estimation.php
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