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May 14th, 2012
Happy May Term, everyone!
For the students in my chess class, here is an awesome illustrated guide to the basic rules of chess. This page shows you how each piece can move. I highly recommend that you bookmark this page and have it handy on your phone/iPad/portable computing device. You can use these notes in class!
If you’re interested in the complete rules of tournament chess, check out this page here.
Meanwhile, everyone – yes you! – should download a chess clock app for their phone/iPad, etc for the rest of the class. I am not going to link to anyone one app because I don’t want to endorse any one app or another. Just get a free one – there is no need to pay for that app!
And if anyone can bring another board or two to class tomorrow, that would be great!
Image credit: Cool chess picture stolen from Encyclopaedia Britannica’s web site.
April 30th, 2012
Forgive me my dated pop-culture references. I just can’t help myself sometimes.
Today we’re going to be talking about the properties of exponents. Now, if you’ve been in my classes for any length of time, you know that I’m not a big fan of memorize and regurgitate. I like to explain why things work the way they do, and I love to give context (especially historical context) to anything I can. But when it comes to things like this, the best thing you can do for yourself and your math grade (present and future) is just memorize the following rules. Anytime you have an expression with exponents, these rules will always apply. Always. No exceptions!
In all of the following examples a and b are a real, non-zero numbers – in other words, they can be positive or negative, a fraction or a decimal, but they cannot be zero or imaginary.
| Property |
Explanation |
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Anything to the zero power is 1 |
 |
Anything to the power of 1 is itself |
 |
A negative exponent flips the number – if it was a whole number, it is moved to the bottom of a fraction; if it’s a fraction, the number gets flipped!
|
 |
You can rewrite the root of number as a fractional exponent |
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You can break the root of a fraction into the root of the numerator over the root of the denominator |
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When you multiply, you add the exponents |
 |
When you divide, you subtract the exponents |
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When you take the power of a power, you multiply the exponents |
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When you take the root of a power, you divide the exponents |
If you go to the effort to memorize all of these properties, you will have a lot fewer problems in algebra (as well as precalculus and calculus). Want some practice using these? Okay, click on this link to download some practice that you can turn in for extra credit. The worksheet is due by 2 p.m. Thursday, May 3. No late papers – no exceptions!
Good luck! And remember, there is still after school math help Monday through Thursday this week and next week. Hey, it’s free!
Image credit: I stole this image of Prince Adam becoming He-Man from a random Google search, and then I went crazy with the Glowing Edges filter in Photoshop. He-Man and the Masters of the Universe are copyright Mattel Inc.
April 26th, 2012
It’s that time of year. The time when final exams loom on the horizon like a flock of birds sitting above a freshly washed car. To help you all prepare for the inevitable mess, we will be spending the next few weeks reviewing the most important algebra 2 concepts of the year. Today, we start with solving equations.
If you’ve paid any attention to any of our class discussions this year, you’ve heard me talk about the perils of doing one thing and saying another. (The problems created by this phenomenon are not limited to the field of mathematics, for the record.) When faced with solving an equation like this:
x2 – 4 = 16
… A fair number of people – you might even be one of them – will look at that and think: “I’ll just take the square root of both sides.” This is a perfectly legal move if you actually do what you said you were going to do. Unfortunately, too many people actually do this:
x – 4 = 4
which is NOT actually taking the square root of both sides. If you just take the square root of x2 and ignore the -4, you’re not taking the square root of that side – you’re just playing favorites and taking the square root of the obviously squared term. And math will make you pay for playing favorites, trust me!
To make matters worse, just as many well-meaning students will do this:
x – 2 = 4
“See?” they seem to say, “I didn’t forget to take the square root of 4!” While it is true that people who do this are, indeed, taking the square root of both critters on the left side of this equation, they are still not taking the square root of that SIDE of the equation.
If you were serious about taking the square root of both sides, you would do this:
Now, you’re finally doing what you said you were going to do. You are finally taking the square root of both SIDES of that equation!
There’s only one small problem at this point. There is no easy-to-find square root of x2 – 4. The square root of that expression would be something of the form (x + a) • (x +a). But there is no real binomial square root of x2 – 4. Which means … we can’t really solve this problem by taking the square root of both sides first. For realz.
Yes. I just made you read all of this only to find out that you actually can’t solve this problem by taking the square root of both sides. But, at least you know how to tell if you’re taking the square root of both sides of an equation properly. That’s something, right?
So, how do you solve it? Well, if you can’t take the square root of both sides – what other things have we done to solve equations over the years? How about adding 4 to both sides. If you do that, you end up with :
x2 = 20
Now if you wanted to take the square root of both sides, it would be easy. The square root of x2 is simply x and the square root of 20 is √20. (Technically, it’s ± √20, but we’ll take more about that later.) Believe it or not – that is the solution.
Why did I make you go through all of this? For three reasons:
1. To reinforce the fact that you sometimes do one thing and say another. This will continue to cause you problems.
2. To demonstrate that just because you first thought didn’t work out, it doesn’t mean there is no solution. You have years of solving equations under your belt, so you always have other options to try.
3. To give you a chance to practice solving equation for extra credit. That’s right! Your reward for reading this far is another review worksheet that you can complete for extra credit. (Click on this link to download the paper.) This paper is due MONDAY, APRIL 30 by 2 p.m. I will not accept ANY papers after 2 p.m. for any reason at all!
Good luck!
Image credits: Right. Well … um, I stole the “Keep Calm” sign from a Google image search. And there was no given attribution for this picture of the classic British World War 2 poster. So … there it is. I suppose I should credit the UK’s Ministry of Information.
April 15th, 2012
Image credit: Me, from my chair on a lovely spring day.
March 29th, 2012
NOTE: The following are excerpts from the Diagnostic and Statistical Manual of Mental Disorders of the American Psychological Association. Before ANYONE thinks they are qualified or justified in using any of the following information to diagnose, accuse, or in any way attempt to help anyone else, please read this Cautionary Statement:
The specified diagnostic criteria for each mental disorder are offered as guidelines for making diagnoses, because it has been demonstrated that the use of such criteria enhances agreement among clinicians and investigators. The proper use of these criteria requires specialized clinical training that provides both a body of knowledge and clinical skills.
In other words, just because you think that someone fits the criteria outlined in any of these diagnoses – that doesn’t make you qualified to make such a pronouncement. So, don’t think that just because you read something online that you can tell anyone else what is wrong with them – or what they need to do to “fix” themselves. This means YOU!
In our AP Psychology class the other day, we were talking about different types of personality disorders and I got a bit confused about three different types of disorders. For your edification (and mine) here are the diagnostic criteria for anti-social personality disorder, narcissistic personality disorder and avoidant personality disorder.
Diagnostic Criteria for 301.7 Antisocial Personality Disorder
A. There is a pervasive pattern of disregard for and violation of the rights of others occurring since age 15 years, as indicated by three (or more) of the following:
(1) failure to conform to social norms with respect to lawful behaviors as indicated by repeatedly performing acts that are grounds for arrest
(2) deceitfulness, as indicated by repeated lying, use of aliases, or conning others for personal profit or pleasure
(3) impulsivity or failure to plan ahead
(4) irritability and aggressiveness, as indicated by repeated physical fights or assaults
(5) reckless disregard for safety of self or others
(6) consistent irresponsibility, as indicated by repeated failure to sustain consistent work behavior or honor financial obligations
(7) lack of remorse, as indicated by being indifferent to or rationalizing having hurt, mistreated, or stolen from another
B. The individual is at least age 18 years
C. There is evidence of Conduct Disorder (see p. 90) with onset before age 15 years
D. The occurrence of antisocial behavior is not exclusively during the course of Schizophrenia or a Manic Episode.
Now, compare those criteria with that of …
Diagnostic Criteria for 301.81 Narcissistic Personality Disorder
A pervasive pattern of grandiosity (in fantasy or behavior), need for admiration, and lack of empathy, beginning in early adulthood and present in a variety of contexts, as indicated by five (or more) of the following:
(1) has a grandiose sense of self-importance (e.g., exaggerates achievements and talents, expects to be recognized as superior without commensurate achievements)
(2) is preoccupied with fantasies of unlimited success, power, brilliance, beauty, or ideal love
(3) believes that he or she is “special” and unique and can only be understood by, or should associated with, other special or high-status people (or institutions)
(4) requires excessive admiration
(5) has a sense of entitlement, i.e., unreasonable expectations of especially favorable treatment or automatic compliance with his or her expectations
(6) is interpersonally exploitative, i.e., takes advantage of others to achieve his or her own ends
(7) lacks empathy: is unwilling to recognize or identify with the feelings and needs of others
(8) is often envious of others or believes that others are envious of him or her
(9) shows arrogant, haughty behaviors or attitudes
When you compare these two disorders, you can see that they have a lot of overlapping behaviors. Whether it’s “lack of remorse” (from anti-social) or “lacks empathy” (from narcissistic), we are still talking about people who – for somewhat different reasons – are not cognizant of others. Outside of a clinical setting, it might be hard to tell if someone is rude to others because he simply has no regard for societal norms (anti-social) or thinks those around him aren’t worth his extra time because they are beneath him (narcissistic). Also, the subject in question may simply be a normal, American teenager. It’s very hard to tell the difference sometimes.
But when a lot of people talk about “anti-social behavior,” what they’re really talking about is behaviors that helps them avoid social situations. In the DSM-IV, this is called …
Diagnostic Criteria for 301.82 Avoidant Personality Disorder
A pervasive pattern of social inhibition, feeling of inadequacy, and hypersensitivity to negative evaluation, beginning by early adulthood and present in a variety of contexts, as indicated by four (or more) of the following:
(1) avoids occupational activities that involve significant interpersonal contact, because of fears of criticism, disapproval, or rejection
(2) is unwilling to get involved with people unless certain of being liked
(3) shows restraint within intimate relationships because of fear of being shamed or ridiculed
(4) is preoccupied with being criticized or rejected in social situations
(5) in inhibited in new interpersonal situations because of feelings of inadequacy
(6) views self as socially inept, personally unappealing, or inferior to others
(7) is unusually reluctant to take personal risks to engage in any new activities because they may prove embarrassing
I hope this clears up a few things.
By the way, if you’d like some insight into what’s cooking for the upcoming DSM-V, read this.
Image credit: A collection of past DSMs, courtesy of the APA.
February 27th, 2012
I really have no idea how I got this all turned around last week in my algebra 2 classes. I double checked with my precalculus students, and I taught them the right way to do composition of functions last semester. I don’t know what was wrong with me last week. I suppose I was just on auto-pilot and wasn’t thinking. Let this be a lesson to you all: Never stop thinking!
So, without any further rationalization, here is the proper way to do Composition of Functions:
A Note about Notation: Our book uses this notation (f ∘ g)(x) … which you need to recognize. Most people I know prefer the following notation: f(g(x)). For the rest of this lesson, we will be using the double parenthesis notation. Onward!
Let us consider the following set of functions (color-coded, for your convenience):
f(x) = 2x – 3
g(x) = 5 – x
h(x) = –x2
If we have the problem of f(g(x)), we will work from the inside out (as I said last week, but didn’t actually do). So, you take the function on the inside of this problem – g(x) in this example – and replace every x in the outside function – f(x) – with whatever g(x) is. I’ve gone ahead and changed the x’s we’re going to replace to green (g for green!). So, given our f(x) …
f(x) = 2x - 3
When we replace the x with g(x), it would now look like this:
f(g(x)) = 2(5 – x) – 3
Following the distributive property, we end up with …
f(g(x)) = 10 – 2x – 3
Combine like terms, and it’s …
f(g(x)) = 7 – 2x
Get it? We substituted every x in our original function – f(x) – with the stuff from g(x). That’s all we do!
Let’s try it with f(h(x)). Again, we’re going to replace every x in our f(x) with whatever h(x) is. So, in this case, the x’s will be purple (because h(x) is purple).
f(x) = 2x - 3
Replace the x with h(x) …
f(h(x)) = 2(–x2) – 3
Distribute the 2 …
f(h(x)) = -2x2 – 3
And that’s it! I hope that clears up any problems that I created. Everything else learned about combining functions – adding, subtracting, multiplying, dividing – that was all correct. So, no need to worry about that. We will be practicing this in class tomorrow, so don’t worry about that. Good luck!
Image credit: I would LOVE to take credit for that most excellent composition of Madness (a group from the 1980s) and functions, but I can’t. I just Googled “function madness” and that was one of the first images that popped up. I stole it from Flickr user Stéphane Massa-Bidal. Genius, sir. Pure genius!
February 22nd, 2012
As promised, here is a quick guide to the neurotransmitters in your brain. If you’re not in AP Psychology, you still might find this interesting. Sort of. Maybe. Kinda.
That is all.
January 25th, 2012
I have often complained that many textbooks explain the relationship between logarithms and exponents in a needlessly confusing way. Here, for example, is how our current textbook explains the general form of an exponential function:
y = abx
Twenty pages later, it discusses the general form of a logarithmic function like this:
logbx = y
And then a few pages later, it gives the change of base formula like this:
logax = logbx / logba
So is a the base sometimes, but the answer others? And why was the x the exponent in the first example, but the answer in the second? And what is b supposed to represent in the second example as opposed to the third? Where’s the consistency? Math people should be bigger fans of consistency in syntax and symbolism. But they’re not. Thus, I spend more time railing at the bad writing of textbook authors than is really healthy. That’s my problem, however. Let’s talk about yours.
I want my students to understand the reciprocal relationship between exponential expressions and logarithmic expressions. To help illustrate that relationship, I like to use the following equation:
bx = a
Where b is the base (the thing being taken to a power), x is the eXponent (the power we’re modifying the base by) and a is the answer (the result of taking b to the x power!). I like these letters in the positions because they are better mnemonic devices: b for base, a for answer, etc. And so, when I want to talk about the logarithm form of the same expression above, I move the same symbols:
log b a = x
What I hope this illustrates is that the answer to any logarithm problem is the exponent. The whole point of having logarithms (at least for our purposes) is to solve an equation where the unknown is the exponent. Here’s an example:
4x = 64
In this case, if you can remember that 4 is the base and 64 is the answer, then to solve this problem you set it up like this:
log 4 64 = x
Not only is x the unknown but, because we’re using my variables, you should remember that the unknown is the exponent. Doesn’t that make more sense? I think it does.
We can even apply this to the change of base formula. Because we use either base 10 or base e (more about that later), there’s no need to bring in a new variable – or, worse, reuse an old variable in a different place. In my world, the change of base formula is written simply as
log b a = log 10 a / log 10 b
In plain English, the change of base formula takes the log of the answer over the log of the base from the original problem. If this doesn’t make sense, leave me a comment. I hope this helps!
January 24th, 2012
My AP Psychology class is looking into the relationship between social network usage (Facebook, Twitter, MySpace and even texting) and sleep in high-school age students. Later in the semester, we will be asking for volunteers to keep track of their social networking and sleep habits. But for right now, we have a lot – and I mean a LOT of background research to do. Which means we have a lot of reading to do.
Here is a list of what we’re reading. The first section are online articles you can read. The second section has PDFs you can download and read at your convenience. If anyone (in or out of my class) has other ideas, feel free to leave links in the comment section.
Oh, and if you’re in my class now and you’re wondering why I didn’t post a link to the article you sent, there are three possible reasons:
1.The age group studied is not the population we’re studying (elementary or college students instead of high school)
2.The behaviors studied are not the behaviors we are examining (eg, drug use is not within the scope of our study)
3.The link you sent me is dead. Feel free to try again.
Online articles
• Adolescents Living the 24/7 Lifestyle: Effects of Caffeine and Technology on Sleep Duration and Daytime Functioning
• Bedtime Activities, Sleep Environment, and Sleep/Wake Patterns of Japanese Elementary School Children
• Use of information and communication technology (ICT) and perceived health in adolescence: The role of sleeping-habits and waking-time tiredness
• How light affects our sleep
• Adolescent Use of Mobile Phones for Calling and for Sending Text Messages After Lights Out: Results from a Prospective Cohort Study with a One-Year Follow-Up
Articles to download
• The landmark Brown University School Sleep Habits Survey. If you read nothing else on this page, check these out to see what we’re up against in this project.
• Sleep Needs, Patterns and Difficulties of Adolescents This is that great general overview article I was talking about in class on Tuesday. This is also a must-read!
• Health Problems with the use of information techonlogies
• Effect of Illuminance and Color Temperature on Lowering of Physiological Activity
• Effect of Color Temperature of Light Sources on Slow-wave Sleep
• The Impact of Media Use on Sleep Patterns and Sleep Disorders among School- Aged Children in China
• Evening exposure to a light emitting diodes (LED)-backlit computer screen affects circadian physiology and cognitive performance
• Television Viewing, Computer Game Playing, and Internet Use and Self-Reported
Time to Bed and Time out of Bed in Secondary-School Children
• Physiological Effects of Sudden Change in Illuminance during Dark-Adapted State
• Sleep patterns, electronic media exposure and daytime sleep-related behaviours among Israeli adolescents
Image credit: The Y U No guy, courtesy of Meme Generator
December 29th, 2011
So, my daughter is (sporadically) into performing magic. She has a few kits, a magic wand and several props in her arsenal already, but she recently came across a trick that requires a special bag. Because we’ve been watching lots of Phineas and Ferb and we have plenty of time on our hands, I suggested we make one. To be clear: The only experience I have with sewing is reattaching buttons to my coat. But, hey – people were sewing before they had language and sanitation, so I figured I could make a simple bag, right?
For once, I was right! Here are a few snaps from the process.
First step, cutting out the pieces.
Trying to make straight seams. I now understand why safety pins and thimbles were invented.
A flip and a tuck and …
Ta da! The finished product
Okay, so taking a picture of someone with Asian skin wearing a pink shirt holding a pink bag sitting in front of a melon-colored wall … not the best color composition. I’m just working with what I’ve got here, people.
Enjoy what’s left of your break!
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