Newton's Method

One application of derivatives it to use the equation of a tangent line to approximate values close to a known value on a curve. i.e. finding √35 on the function f(x) = √x.

Another use of derivatives is to find a tangent line whose root is the same as the root of a more complicated funtion. Every approimation we make gets us closer to the actual root of the more complicated curve. This is known as Newton's Method and it's named after the man who first developed the technique, me! No, just kidding, Sir Isaac Newton developed the technique. Watch how it works here and here.

Here is a flash tutorial that explains the process we discussed in class today. You can watch Newton's Methis in action using this example. And then try these exercises to see how well you can apply what you've learned; detailed solutions are provided.

Derivative Acrostics

Kudos to Mrs. Armstrong for turning me on to the idea of acrostics in math.

Blogging Prompt
Your task is to create an acrostic "poem" that demonstrates an understanding of calculus related to any one of these concepts:

DERIVATIVE
POWER RULE
PRODUCT
QUOTIENT
CHAIN RULE
TANGENT LINE
NEWTON'S METHOD

As an extra challenge (worth an additional bonus mark) try to make a Double Acrostic, that is, each line should begin and end with a letter of the word you are working with.

Remember, this is a bit of a race. Your answers have to be posted to the blog in the comments to this post. If someone has already used a word or phrase in their acrostic you cannot use the same word or phrase. i.e. It gets harder to do the longer you wait. ;-)

Here is an example of an acrostic that Mrs. Armstrong wrote:

Always in 2 dimensions
Region between the boundaries
Entire surface is calculated
Answer is in units²

Be creative and have fun with this!!

Sunday Connected-Slide Funday!

ConSlide puzzles are a new type of sliding block puzzles invented by M. Oskar van Deventer... The pieces of a ConSlide puzzle move like regular sliding block puzzles, but some of the pieces have sections connected by bars of various heights. This means pieces can pass over and under one another as long as the bars and posts don't run into one another. The goal of each of these puzzles is to move the red block to the upper left corner. Play it here.

(Once again, thanks to Think Again!)

The Editor's Initiative

Instead of posting a pre-test reflective comment on your progress in this class you may undertake The Editors' Initiative. Here's how it works:

  Step 1: Scan through the previously posted Scribe Posts on the blog. Find one that has one or more errors.

  Step 2: Discuss the error(s) and what you think the correction(s) should be with me. If I agree with your editorial proposal go to Step 3.

  Step 3: Discuss the editorial change with the author of the post. The author will chose to proceed in one of the following two ways.

3a	3b
The Editor is briefly allowed administrative privileges on the blog. They will edit the post to make any necessary corrections. They then sign the post at the bottom: Edited by: [name] on [date]	The author will edit the post in consultation with the editor who will vet the author's changes until they are correct. The author then signs the post at the bottom: Consulted editor [editor's name] on [date]

Students may chose to make more than one edit. Each additional edit will earn them a bonus mark on the next test. Your mark on the previous test determines the maximum number of edits/bonus marks available to you.

Mark on Last Test / Max Edits Allowed
> 90 / 1

80-89 / 2 (1 bonus mark)

70-79 / 3 (2 bonus marks)

60-69 / 4 (3 bonus marks)

50-59 / 5 (4 bonus marks)

40-49 / 6 (5 bonus marks)

30-39 / 7 (6 bonus marks)

20-29 / 8 (7 bonus marks)

10-19 / 9 (8 bonus marks)

0-9 / 10 (9 bonus marks)

You may also assume the role of Content Consultant to earn marks as outlined above. Here's how it works:

  Step 1: Scan through the previously posted Scribe Posts on the blog. Find one that doesn't provide enough detail or leaves out too much information. Decide what additional content should be added.

  Step 2: Discuss the new content you think should be added with me. If I agree with your editorial proposal go to Step 3.

  Step 3: Discuss the editorial change with the author of the post. Together, you will chose to proceed in one of the following two ways.

3a	3b
The Content Consultant will add a new post to the blog inserted at the appropriate time and date. They then sign the post at the bottom: Additional Content by: [name] on [date]	The author will edit the post to include the additional content provided by the consultant. Additional content will appear under a heading "Additional Content". The author then signs the post at the bottom: Additional Content Provided by [consultant's name] on [date]

Students may chose to make several additional content contributions for bonus marks according to the table above.

As we discussed in class, you can edit your own scribe posts only if the whole class agrees with your proposal and we are convinced that you tried to do your best work the first time around. When it's your turn to be scribe try to write a post that is so excellent no will be able to edit it. ;-)

Mr. Dixon's on eBay - How Trusted is He?

Mr. Dixon is in my classroom right now and he has a question he needs your help with.

His trust rating on eBay is 99.2%. Out of 122 transactions he's had, 121 people said he treated them well and he can be trusted.

Mr. Dixon wants to know when will his "trust rating" reach 99.3% and he needs your help.

(1) Can you write an equation he can use to solve this problem?
(2) Can you solve the equation and tell Mr. Dixon how many more favourable exchanges he needs to have to raise his trust rating to 99.3%.

All of my classes are working on this. The first student in each class who solves Mr. Dixon's problem correctly gets a chocolate bar. ;-)

Related Rates Homework

Read through this tutorial on related rates problems. It's a lot to read but it's worth your time. The beginnig will take you through the general steps to take in solving related rates problems, then threre are several examples that have animations to show you what is changing in the problem and how.

Here is another tutorial with animations and a problem set. You can do as many questions from the problem set as you like. But your assignment is to do the first 5 questions here. Answers are provided so see if you can solve them without looking at the answers. ;-)

Fourth Powers

- I hear you have extraordinary powers!

- Not true! What I said to the press was that I have fourth powers at home.

- What do you mean?

- 1, 16, 81, .... That kind of stuff.

- But these are square numbers!?

- Yes, but they are more than that. They are square numbers squared.

- I see. And you have these at home?

- Some of them. Not all of them.

- And what do you do with them?

- Play with them, of course, what else?

- What kind of games?

- This morning I added them.

- That must have been great fun!

- That's what I said to the press, but they didn't believe me.

- How did the game go?

- Not very well. I tried to pick less than nineteen of them, one several times if
neccessary, to get any number in the universe.

- How did you get 20?

- 20 was easy. 1 + 1 + 1 + 16 = 20. I only had to use five of them.

- Which number gave you problems?

- I found a number less than 500 that required nineteen fourth powers.

- So you lost the game?

- Yes, and I can't even remember which number it was.

The View From Sunday

This week's puzzle comes from Think Again! again. ;-)

The Other View


With nine pieces as the one shown above, a 3x3x3 cube was made. Below you can see the cube viewed from South, East, and North. What does it look like seen from the West?



Say .... uh .... you've already finished your homework right? The stuff three posts below?

Visitors From Alabama

This was left as a comment to Jayson in his pre-test reflective post. I thought I'd bring it out here for all of you to see .... our class has just grown. ;-)

I just want you to know that I am an AP Calculus AB teacher in Birmingham, Alabama, and I was just introduced to your class' blogs yesterday during a workshop. I think that this is a really neat and enriching idea to do in a mathematics class. I am going to try to learn how to get this set up for my classes, but until someone has enough time and energy to train me, I am going to require that my students visit, learn from, and comment on your class blogs. I really respect this particular blog of yours, and I will read it first thing Monday morning to all of my classes. I hope that you all and Mr. K keep up the incredible work!!!
Mrs. B from Alabama

del.icio.us Guide

You can download a pdf guide to del.icio.us right here.

It is 12 pages long. Flip to the section where you feel you most need help. If you like to work off a hard copy you can print it up. If you don't have a printer at home you can print it up at school. If enough of you want to have the hard copy let me know in class and I'll have a bunch of them copied for you. ;-)

Don't forget to do your homework in the post below this one!

Product and Quotient Rule Homework

As promised in class, your homework this weekend is online.

We'll begin with this combination quiz and review of the product and quotient rules. The page is a little long but make certain you work through ALL the examples and exercises.

Sarah continues to do excellent work on her blog. Part of your homework this weekend is to visit her blog and work through the first 4 movie links. You can also watch the Chain Rule movie to get a preview of what we're doing next week.

Then you can take this quiz. Actually there are two quizzes on this page, you only have to do quiz 1 questions number 1 through 6. The other questions involve derivatives of the natural logarithm function and composites of functions. We'll be covering these topics next week -- so you can come back to this quiz then.

Lastly, take this quiz on the product and quotient rules. There are twelve questions -- you have to do the first eleven. If you watch the video over at Sarah's blog on the chain rule you'll be able to do the twelfth question as well. You'll come across some notation you're not familiar with in this quiz in 1 or 2 questions. You can click on the answer link to learn how it's used and what it means.

Have a great weekend!

A del.icio.us Idea

I recently received this email from a student in another class:

Hey Mr. K.

This is one of the websites I was looking at that had simplifying radicals..

http://regentsprep.org/Regents/math/radicals/pracRad.htm

I found a few that I thought were good just by typing "radicals" in google, they really helped me out.

See you Monday,
************

Students often find more, and better, sites than I do. You're better websurfers than I am. ;-) That got me thinking .....

I spend a lot of time looking for good websites that help us learn in this class. But what if we all spent a little time doing that? What if there was an easy way for us to both save our bookmarks (without cluttering up our favourites list) and share them with the whole class with the click of a single button? And what if we could access those bookmarks not just from home, but from any computer in the world? Hmmm .....

Well, there is an easy way to do that! Instead of saving bookmarks on your home computer sign up for a free account at a site called del.icio.us. You can then access them from any computer in the world. You can easily install a little button/bookmark that allows you to save any webpage you're looking at without interupting your surfing. Tag it using this tag:

apcalc

Now we can all get each others bookmarks with the click of a single button in our del.icio.us accounts.

I'll go one better than that. If you all jump in on this idea, I'll write a post on our blog (with a permanent link in the sidebar) that will load the 10 most recently saved links automatically as you find them. I'll also include a link to the archive that you can browse at your leisure.

If this interests you (and I can't imagine how it doesn't) read this tutorial on how to get started with del.icio.us. You might also be interested in watching this screencast that illustrates just how powerful this web tool is.

When you've signed up for a del.icio.us account (register here) leave a comment on this post telling me so. When I see some action here I'll blog that self-updating post. ;-)

It's Sunday Again!

In the game of Kayles a number of pins (or coins) are arranged in separate rows. A legal move consists of knocking down either one pin or two adjacent pins from the same row. This may break up the row into two smaller rows. Whichever player knocks down the last pin wins. What's the winning strategy for the game of Kayles?

Play here. Discuss your winning strategies in the comments to this post .... and have fun doing it!

Another Feather in Your Cap

About a month ago I was contacted by Meg Gwaltney who works for Stein Communications, an education marketing firm in the US. She had stumbled across our blog and was impressed with our work. So impressed she wrote an article about us that is distributed across the United States. You can read it here.

You continue to make an impact. Have you decided how you're going to move this to the next level?