This is a post from the most interesting math blog I have ever found (POLYMATHEMATICS).

Every year I get a few kids in my classes who argue with me on this.  And there are arguers all over the web.  And I just know I’m going to get contentious “but it just can’t be true” whiners in my comments.  But I feel obliged to step into this fray.

.9 repeating equals one.  In other words, .9999999… is the same number as 1.  They’re 2 different ways of writing the same number.  Kind of like 1.5, 1 1/2, 3/2, and 99/66.  All the same.  I know some of you still don’t believe me, so let me say it loudly:

Do you believe it yet?  Well, I do have a couple of arguments besides mere size.  Let’s look at some reasons why it’s true.  Then we’ll look at some reasons why it’s not false, which is something different entirely.  The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this.  Let x = .9999999…, and then multiply both sides by 10, so you get 10x = 9.9999999… because multiplying by 10 just moves the decimal point to the right.  Then stack those two equations and subtract them (this is a legal move because you’re subtracting the same quantity from the left side, where it’s called x, as from the right, where it’s called .9999999…, but they’re the same because they’re equal.  We said so, remember?):

Surely if 9x = 9, then x = 1.  But since x also equals .9999999… we get that .9999999… = 1.  The algebra is impeccable.

But I know that this is unconvincing to many people.  So here’s another argument.  Most people who have trouble with this fact oddly don’t have trouble with the fact that 1/3 = .3333333… .  Well, consider the following addition of equations then:

This seems simplistic, but it’s very, very convincing, isn’t it?  Or try it with some other denominator:

Which works out very nicely.  Or even:

It will work for any two fractions that have a repeating decimal representation and that add up to 1.

Those are my first two demonstrations that our fact is true (the last one is at the end).  But then the whiners start in about all the reasons they think it’s false.  So here’s why it’s not false:

• “.9 repeating doesn’t equal 1, it gets closer and closer to 1.”

May I remind you that .9 repeating is a number.  That means it has it’s place on the number line somewhere.  Which means that it’s not “getting” anywhere.  It doesn’t move.  It either equals 1 or it doesn’t (it does of course), but it doesn’t “get” closer to 1.

• “.9 repeating is obviously less than 1.”

Hmmmm…it might be obvious to you, but it’s not obvious to me.  Is it really less than 1?  How much less than 1?  No, seriously…tell me how much less?  What is 1 minus .99999999…?

Really???? Infinitely many zeros and then after the infinite list that never ends, there’s a 1????  Surely that’s stranger than the possibility that .9 repeating simply does equal 1.  Or for something even stranger, consider this:  if .9 repeating is less than 1, then we ought to be able to do something very simple with those two numbers:  find their average.  What’s the number directly between the two?  Or for that matter, name any number between the two.  Let me guess:  the average is .99999…05?  So after this infinite list of 9s, there’s the possibility of starting up multiple-digit extensions?  Doesn’t that just raise the obvious question:  What about .9999999…9999999…?  Namely, infinitely many 9s, and then after that infinite list, there’s another infinite list of 9s?  How, exactly is that different from the original infinite list of 9s?  If you saw it written out, where would the break between the lists be?

I’m afraid that if you apply the “huh??” test of strangeness, you get a much higher strangeness factor if you say that .9999999… is not 1 than you do if you say it is 1.

• “Uhhhhh, I’m sorry, but I still don’t believe you.  .99999… just can’t equal 1.”

Well, let’s look a little more carefully at what we really mean by .999999…:

This equation isn’t really up for debate, right?  It’s simply the meaning of our place value system made explicit.  That thing on the right hand side is called an infinite geometic series.  They have been studied extensively in math.  The word “geometric” means that each term of the series is the identical multiple (in this case 1/10) of the previous term. The definition of the sum of an infinite geometric series (and other series, too, but we won’t get into those) goes something this:

1. Start making a list of partial sums:  the sum of the first one number, then the sum of the first two numbers, then the sum of the first three, etc.
2. Examine your list closely.  In this case the list is: .9, .99, .999, .9999, …. (Note that the actual number .99999…. is not on the list, since every number on the list has finitely many 9s.)
3. Find some numbers that are bigger than every single number on your list.  Like 53, 3.14, and a million.
4. Of all the numbers that are bigger than every number on your list, find the smallest possible such number.  I think we can all agree that this smallest number is 1.
5. That smallest number that can’t be exceeded by anything on the list is the definition of the sum of the geometric series.

Notice that I keep putting the word definition in bold face.  (See, I did it again!)  That’s because it’s a definition, which isn’t really up for debate.  It is the nature of a mathematical definition that once you acccept it, you have to agree to its consequences.  In other words, .99999… = 1 by the definition of the sum of a geometric series.  It’s also true if you use the popular formula

a/(1 - r) with a = 9/10, and r = 1/10.

We’re left with this:  merely saying “.99999… doesn’t equal 1″ admits the fact that this number .99999… exists.  And if it exists, it equals 1 by definition.  The only way out for you now, if you still don’t believe it, is to have a different working definition of the sum of an infinite series (go talk to some math professors, and see how far you get) or to deny the very existence of the number .9999….  I have seen a lot of people doubt that the number equals 1, but very few of them are willing to deny the very existence of that number.  If you want to play “there’s no such thing as infinitely long decimal representations,” I’m afraid you won’t get very far, because there’s always the number pi to worry about, too, you know.

Okay, so there’s my rant.  .9 repeating equals one.  No, I’m sorry, it does.

I have been thinking a lot about design as of late.  Being one of the newspaper sponsors at my school, I think about it often.  I have noticed a number of interesting infographics being published lately that consolidate research and statistics into a visual format.

Is teaching design is important for language arts.  I think it is.  Visualizations persuade.  We can convey so much with visual elements.  The lack of words can be powerful as well. (See the example below.)

There are obvious uses for teaching visual rhetoric via film studies (by that I mean filming techniques to foster visualization while reading). Infographics could be used to outline a research project or to be used in a speech.  Anyway, just thought I would share.

Created by Online Education

I recently discovered the Naked Scientists Radio Show podcast. (I found it on iTunes HERE, but the link provided is to the website.)

One of the episodes featured a story about how we can estimate when the story took placed based on the astronomical references. It was just another way to show the possible connections between math, science, and literature. I have (Here is the iTunes link–you’ll find the story at the 6:00 mark). The link and description below are from the results of searching for Odyssey on their website.

There is another episode that mentions The Odyssey listed below as well. The link is to the website, but here is a link to the iTunes podcast where you will find the story at the 21:09 mark.

I hope you will find these useful.

		Ancient Poet Astronomically Accurate
	Ancient Poet Astronomically Accurate , , A few weeks ago on the Naked Scientists we followed the story of the Odyssey, and how although the land has changed in the 3000 years since it was written, the poet knew his geography. Parts of the poem allowed modern …
	Size: 2.1 K - Created: 01-07-08 - Modified: 01-07-08 17:21

		The Secrets of Odysseus
	… of the most famous of those ancient Greek myths, the Odyssey, might actually be based on a lot more fact than … of the most famous of those ancient Greek myths, the Odyssey, might actually be based on a lot more fact than … of the most famous of those ancient Greek myths, the Odyssey, might actually be based on a lot more fact than … stand these two great epic poems, the Iliad and the
	Size: 14.9 K - Created: 10-06-08 - Modified: 11-06-08 21:09

I discovered an new website today called cohere. It is Twitter with a slice of Bubbl.us on your favorite news website bun. This is another example of how emergence could be used to bring attention to important topics/information.

Check out the video below for more info.

About Cohere

We experience the information ocean as streams of media fragments, flowing past us in every modality.

To make sense of these, learners, researchers and analysts must organize them into coherent patterns.

Cohere is an idea management tool for you to annotate URLs with ideas, and weave meaningful connections between ideas for personal, team or social use.

Key Features

• Annotate a URL with any number of Ideas, or vice-versa.
• Visualize your network as it grows
• Make connections between your Ideas, or Ideas that anyone else has made public or shared with you via a common Group
• Use Groups to organise your Ideas and Connections by project, and to manage access-rights
• Import your data as RSS feeds (eg. bookmarks or blog posts), to convert them to Ideas, ready for connecting
• Use the RESTful API services to query, edit and mashup data from other tools

When I was in college I strolled into the bookstore and picked up a notebook for my class and headed to the classroom. Once the professor entered I flipped it open to find that the first page was defective. The margin line ( which is usually just an inch or so from the left side of the page) was about 4 inches from the left (as seen in the picture to the right). I flipped through the notebook to find that the whole thing was defective. Now planning to return it, I closed the it. In doing so I discovered that I hadn’t bought a college ruled notebook; rather, I had purchased a law ruled notebook.

Over the semester, I found that it was the greatest notebook ever! I found it to be very useful for organizing notes. (To experience law ruled paper, just print it from pritablepaper.net (which is explained in this post).

FOR LITERATURE: To the left of the red line I would write character names, page numbers, or dates. To the right of the red line I would write details from the text, professor commentary, or personal commentary.

FOR PAPERS: I would write my drafts of paragraphs or of outlines to the right of the red line. To the left I would write notes to myself for changes. (I know you can do that on typical lined paper, but with this paper you have more room to make more significant/substantial changes.)

FOR MATH: Because I am a why guy (I need to know why things work the way they work), I needed to take notes on problems demonstrated in class to the right of the red line and explain to myself why I did certain operations to the left of the red line. This helped me when I was studying in the wee ours of the morning after the logic portion of my brain had gone to sleep.

OTHER USES:

• Vocabulary study guide (just fold it back and your have a study tool).
• Words and definitions
• A page to keep up with assignments you’ve had throughout the semester.
• Have your students take tests on it so you can comment on their performance.
• Since I am sure you can come up with a million more uses, I will stop the list here.

Please feel free to leave other suggestions in the comments below.

Our 27th (of 214) posts was an article by Chris about twittering your lesson plans.  Recently, while playing around posterous, I realized how easy it would be to simply send an email to a blog post (or twitter) a couple of questions from the day’s discussions in order to store up questions for your exam.

I will start doing this as soon as I return from our break.  I will not give my students my posterous blog’s name because they should be taking notes.  However, as I tag my blog posts, I will have a record of topics they should revisit for the exam.

The beauty of this idea is that I could either post the actual information they need to know (like the rules of commas) to direct them to the blog around exam time.  Or I could write out the actual questions out and set the blog to private.

The reason I am so interested in posterous is because you can simply send an email from your computer or text message the question from your phone without ever logging in at posterous.com.

TO SEE MORE POSTS ABOUT POSTEROUS, CLICK HERE.

The other day I discovered POSTEROUS. It is a blog site that allows you to simply email your posts right to your blog. That way you NEVER have to sign in and deal with all of the passwords. All you do is drop whatever you want in your post into an email and send that puppy!

It is so easy to embed any of the following, which would typically require access to other sites to get embed code to insert things like:
Word Documents
PowerPoints [sic]
Pictures
ExcelDocuments
So on and so forth…

It says you do not have to create an account, but I would suggest doing so. That will allow you to have your own original name for your blog. It only takes a second. In fact, it is easier than setting up an email account.

Using Posterous for a classroom blogging community:

Before I wrote that I would not suggest using this for your classroom blogging community, but I have changed my mind. As you can see in my comments, Garry [sic] Tan one of the founders of that site has reported that there will soon be password protected groups. This will make this blogging community even more secure than blogger or blogspot. That’s AWESOME! Thanks for reporting back, Garry.

If there were settings that would allow my students’ blogs to be completely hidden, I would be sold on using this in my classroom. Any comment on that, Garry? (Garry promptly replied to this question as a comment below.  I think you will like his answer.)  

I would stick to blogger.com or blogspot.com as suggested in this post.

For more directions about establishing classroom blogging communities, see the first part of this post.

Here are the settings I had my students use for their blogs.  These settings apply to blogs created on blogspot.com  (a.k.a. blogger.com).

Once you have created your blog, follow the settings instructions below.  Be sure to follow them exactly.   Your grade will depend on it.

1. click CUSTOMIZE
2. Then click the SETTINGS tab.
3. AND FOLLOW THE DIRECTIONS BELOW EXACTLY.

Click BASIC under the SETTINGS tab.

1. Type in the TITLE of your blog. (Remember, your name cannot appear ANYWHERE on this blog.)
2. Next to COMMENTS click SHOW.
3. Answer NO to all of the questions except these:
   1. Show Quick Editing on your Blog?
   2. Show Email Post links?
   3. Show Compose Mode for all your blogs?
4. Then click SAVE SETTINGS.

Click COMMENTS under the SETTINGS tab

1. Select the option to SHOW your comments. (Sometimes I will post a grade directly to your blog entry, so you want your grade to remain private.)
2. Select the option to give Users with Google Accounts the ability to comment.
3. Select the option to EMBED COMMENTS BELOW POST.
4. HIDE backlinks.
5. Scroll down to the “Comment Moderation” option and select ALWAYS.
6. Next to “Show word verification for comments?” select YES.
7. Next to “Show profile images on comments?” select NO.
8. Then, in the box labeled “Comment Notification Email,” type in your email address.
9. And click SAVE SETTINGS.

Part 2 of this blog can be found here.

Let me begin by saying that doing this is time consuming. It will require about 10 hours of your own time. However, you will quickly see that it save so much more time than that once implemented.

Over the past few weeks I have been working on getting all of my students (98 in all) blogging. There have been inquiries into student access to computers; and there have been hours devoted to figuring out how I would manage all of these blogs; but most importantly, there have been sleepless nights pondering the safety of my students. I started by having parents sign a form giving permission for their student to create a blog. Then I created a blog of my own. Then I walked them through both making their own and adding my blog to their reading list. You can read more about these steps below.

First, I created my form:

This form (as you can see when you click on it) explains how to set a blog up in the first place. I did that so that my students who know their way around the keyboard would go ahead and create theirs. Then, they become my helpers in the classroom. And, as you have probably found, it doesn’t really matter how computer savvy the teacher is. Once you show them something, they take it and make it ten times better.

I gave them a week to bring the signed permission slips back to me. Then we spent the week with the computers. However, each day I had a literature-relevant prompt to get them writing.

IT IS VERY IMPORTANT THAT YOUR SUBMERGE YOUR STUDENTS IN THE BLOGGING PROCESS AT THE BEGINNING.
Here is why:
1.) They are more likely to remember their passwords later if they are submerged in it.
2.) They become attached to it because they get time to personalize it.
3.) They immediately begin getting feedback from their peers.
4.) Then they immediately realized that they are writing for an audience that expects quality writing.
5.) All the kinks will get worked out in the submersion period (I will list a few I discovered below).

IT IS IMPORTANT TO LET THEM HAVE THEIR OWN VOICE
Give them content relevant prompts, but also allow them to express themselves about other topics. (You can see a list of prompts for younger students and another set for older students in the links embedded in this sentence.)

IT IS IMPORTANT THAT THEY EMAIL YOU THEIR USERNAME AND BLOG URL
Be sure to do this. When you have all of their emails, it save a lot of time. I went through and “followed” my students’ blogs once they followed mine. However, because they couldn’t use their real names, I didn’t always know who each blog belonged to until they each emailed me their URL and username. Then, I went to the dashboard of my google reader (which comes with the blogs made at blogger.com) and searched for each students’ blog address. Once I found it, I quickly changed their username as it appeared in my google reader to their real name. That helps when you are trying to give a grade for blog posts.

IT IS IMPORTANT THEY ARE ENCOURAGED (FORCED) TO INTERACT
Forcing them to comment on other students’ blogs will open the dialogue they need to 1.) feel like they are writers, 2.) know that they have an audience, and 3.) open dialogue that forces them to use academic language in “regular” conversation. They would not use this type of language with friends on facebook, but they would use it in the typical college classroom or English class.

ESTABLISH RULES OF ENGAGEMENT
Make sure you have a discussion about appropriate conversations via the internet. I know I have had a number of near altercations from students who were quick to whip out unacceptable commentary in emails to me. Expect that they will do that to their peers as well because it will happen if you don’t talk about it from the start.

MAKE THEM ACCOUNTABLE FOR THEIR POSTS
*BE SURE to have them mark the setting for comments that force the owner of the blog to screen the comments. Then it is the owner’s fault if something inappropriate gets posted. That will make each blogger your front line of defense.
*If you don’t have an iphone or blackberry, and you don’t want to grade while walking the halls, then you can have your students print out their posts (with their comments) to turn in to you.
*Tell them that they must post by a certain date and time for it to count. Most students will be able to post from their cell phones, so they can do it on the bus or math class (no offense, math teachers).

To create an account via blogger (or blogspot), students need their own email address. If they already have an gmail account, they are good to go. If they have their own email address (that is not a gmail account) they CAN just use that.If they need an email account, but you worry about them having their own, check out this post written a while back by Kevin (the post is also referenced in the comments below). However, you may find that guerillamail.com may not work with some blogging sites.

This morning I stumbled across an interesting site that showcased a fun little study guide that anyone can build in about an hour. The plus side to this is that you get to teach/learn a little about circuitry in the process. For students who are a little more interested in science, this might be a way to get them to become interested in vocabulary, math or history.

You could even do this with:
grammar rules
vocabulary
chronology in stories
character traits
chemistry: elements
formulas
science vocabulary
science facts
math equations
math formulas
history time lines
history definitions
historic figures

Check it out:

This week’s project combines some rudimentary circuitry and any subject your child needs a little extra practice in. I would file this with my “Better Than Worksheets” instructional series on drill and practice had I ever created such a series.

The word circuit is obviously historically related to the word circle. Webster’s 1828 defines it thus:

The act of moving or passing round; as the periodical circuit of the earth round the sun, or of the moon round the earth.

Modern technology may have brought some more specific application to the world, but the meaning has not changed much. Circuitry allows electricity to travel around in a circle to do work. Here, we are going to make a simple circuit board that can serve to allow your child to practice any skill that can be answered in a yes/no or multiple choice format. In this case, it will be multiplying by four.

Materials:

file folder (with the open edge trimmed so both sides are th same size and shape)
hole punch
marker
masking tape
aluminum foil
circuit tester

1. Punch holes. You will need two rows of holes. One for each question and one for each answer.

2. Write the problems along one side and the possible answers on the other.

3. Open the folder. On the back side of the side you wrote the problems and solutions, lay a strip of foil between a problem and the correct answer. This is the basis of your circuit board. Fold the ends over the hole and make sure the hole is completely covered by the foil. Insulate with masking tape. Make sure none of the foil is showing on the back side.

4. Continue until all the answers are connected to their problems by a strip of aluminum foil insulated by masking tape. The back will look kind of messy, but that is ok. No one will see it, anyway.

5. Close the file folder. We do a set of problems on each flap, make the circuit board and then tape the folder closed. This helps protect the work.

6. Use the circuit tester to work the problems:

Oops. The circuit was not completed, so the tester did not light up. Try again!

Yeah! She got the problem right, the circuit was completed and the tester lit up.

Caution: When you look for a circuit tester, some contain lead. These are not intended as children’s toys. We searched and found one without lead, but still require our daughter to wash her hands after using it.

THIS POST WAS TAKEN FROM PRINCIPLEDDISCOVERY.COM, A SITE ABOUT HOMESCHOOLING.

Because we are so used to the internet for research purposes, we can spot a cheesy website instantly. MIDDLESPOT.COM gives the option of seeing 50 pages at the same time. In addition to seeing the picture of the site, the site also allows you to see the chunk of text that contains the word(s) you are searching.

My favorite option is that you can add your research to a “wordpad” at the bottom of the page, that way you can pick what you want out of the set of 50 at first, then go back and do a more thorough review to see which ones will make the final cut.

In an effort to raise the bar on my tests, I have sought to make my questions more like those found on AP tests.  I want my students to learn in preparation for the tests, but I also want them to learn FROM the tests.  However, the problem I am experiencing is a lack of knowledge of academic vocabulary.

ACADEMIC VOCABULARY
By academic vocabulary I do not mean words you would likely encounter at each grade level (which is mostly what I get when I google “academic vocabulary”). Rather, I mean the vocabulary test-makers use to craft the questions. Those verbs are what often trip students on such assessments. Since they were difficult to find, I decided to add some of the good stuff I found here on this page.

This list comes from the English Companion (by Jim Burke). It is a list of 350+/- words that one might encounter on an assessment. (Even teachers looking for new words to use in objectives on their lesson plans should check these out.) I don’t think it would be a problem to start off 1st graders on many of these words. What a great list! Thanks Jim Burke!

Test Practice sites (worth your time and for all content areas):
The College Board, which created the AP curriculum, I think, provides some well written free response questions that could easily be adapted for the lower level grades.

Here is a GREAT site for teaching students how to preform at the AP and Pre-AP levels. It has a lot of great links. I will definitely be putting this on my bookmark toolbar!

Of course, I cannot leave out WebEnglishTeacher.com out of the mix. At this link you will find ALL things English. This particular link just deals with AP, but just check out their home page for more great materials on just about anything.

For those looking for practice tests on a variety of subjects, you should check out THIS site. By looking at it, it was created a long time ago, but if you click enough times, you will find some really useful test practice materials. Again, I will say that half of the 100 or so sites do not work, but those that do seem pretty good.
This is for the writing assessment used in Florida called the FCAT (Otownteacher, this might interest you). Click on the drop down menu at the BOTTOM of the page to get a curriculum teaching elementary students how to break down the prompts. The test practice is broken down into weeks. After looking at them, I have found that they would be applicable to all states.