Math: simplifying decimal radicals

Tutoring high school math, you see this a few times each semester.  The tutor shows an example.

Suppose you get asked a question like the following:

Example: Simplify \sqrt{0.225}

Unless you have one of those new-style calculators, like perhaps the WriteView or the natural display, you might likely receive just a decimal.

If you need exact form, here’s how to do it by hand:

1. Rewrite the question in fraction form:

    \[\sqrt{0.225}=\sqrt{\frac{225}{1000}}\]

2. Reduce the fraction inside:

    \[\sqrt{\frac{225}{1000}}=\sqrt{\frac{45}{200}}=\sqrt{\frac{9}{40}}\]

3. Write the radical separately top and bottom (which doesn’t change the value, but can make the problem simpler to consider)

    \[\frac{\sqrt{9}}{\sqrt{40}}\]

4. Where possible, factor each radical so that one part is rootable:

    \[\frac{\sqrt{9}}{\sqrt{4}\sqrt{10}}\]

5. Take the square root where possible:

    \[\frac{3}{2\sqrt{10}}\]

6. Rationalize the denominator; in this case, multiply top and bottom by \sqrt{10} to remove the radical from the denominator.

    \[\frac{3}{2\sqrt{10}}\times \frac{\sqrt{10}}{\sqrt{10}}\]

Since \sqrt{10}\times\sqrt{10}=10, the expression becomes

    \[\frac{3\sqrt{10}}{20}\]

Some exams have “no calculator” sections. One can only wonder what might be on there….

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Linux: Why I use Ubuntu

Tutoring computer science, you need to be aware of operating systems.  The tutor uses Windows, but also Linux – specifically, Ubuntu.

Linux is, as I understand it, a free, open-source operating system.  I got my Ubuntu off the net for free, burned it onto a CD Rom, and booted it up.  It opened a whole new world.

To me, it’s not (only) the free price that makes Ubuntu attractive.  I also have two Windows 7 computers, plus an old XP one.  I don’t know what the Windows operating system actually costs, because the way I get a new Windows system is that I buy a new computer.  For how much we use our computers, I think Windows is a pretty good deal.

My computers that run Ubuntu Linux are over ten years old.  I don’t think they have the resources to run Windows 7 smoothly; they had a hard time running Vista.  Yet, they run Ubuntu like a top.

In my experience, Ubuntu comes with Perl built right in. It also comes with the gcc c compiler.  Loading another compiler is, in many cases, a one-line command.  So to a programmer, Ubuntu is very convenient.

There are some tasks that still seem easier with Windows.  Some hardware is more difficult to run from a Linux system.  My wife only uses Windows or Mac.

For me, the built-in programming facilities – plus the availability of numerous other ones with a simple command – is very attractive.  For that reason, I really like Ubuntu Linux.

I’ll be discussing more about Linux in coming posts:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl: sleep() and the bell

To the tutor’s mind, Perl code is so easy to write, with so many neat functions, it’s a great teaching language.  Today we’ll have some fun.

I’ve heard that some people don’t like programming in Perl.  I just can’t see why not.  I admit, I’m not a professional programmer.  However, for my everyday purposes, Perl is the easiest language.  Apparently, there are some things outside its scope;  in such cases I have to use another language.  When I can use Perl, though, I’m instantly more confident.

I suspect Perl’s lack of rules bothers some people.  After all, Perl’s slogan is “There’s more than one way to do it.”  People who love Perl, probably love it for that same reason.

Anyway, when you know exactly what you want to do, and it’s within Perl’s scope, you can often just write a few lines and come up with something surprisingly functional.  Or, surprisingly neat and fun, in this case.

I discovered Perl’s sleep() function yesterday on stackoverflow.com. It’s straightforward: to pause your program, for example, for 15 seconds, you use the command sleep(15).

What about the bell? It’s ‘\a’, meaning “alert”, and is (maybe) the sound you hear when your computer boots. It’s a throwback to the old typewriter days. Back then, the typewriter would “beep” near the end of the line, alerting that you’d soon go off the paper if you didn’t return the carriage. (I’m really showing my age.)

So here’s a little Perl timer which allows you to set the number of seconds before you hear “beep”:


#/usr/bin/perl

print “Hello:) How long before you want to hear three beeps?\n”;
print “Enter the number of seconds, please.”;
$secs=<STDIN>;
sleep($secs);
print “\n\nHere’s the bell: \a\a\a \n”;

I tried this on my Windows and Linux systems. Sadly, I don’t hear the bell on my flavour of Linux (although it might work on yours:)). It does work well on Windows, though; in fact, it can be quite loud. You may want to adjust your volume downward before running the program:)

I’ll be looking deeper into the issue of why the bell doesn’t work (so far) on my flavour of Linux. My findings I’ll share in a coming post:)

Sources:

Robert’s Perl tutorial

McGrath, Mike. Perl in easy steps. Southam: Computer Step, 2004.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Physics: the wavelength of “middle” F

Tutoring physics 11, this topic comes up most years.  The tutor introduces the formula that relates speed to wavelength and frequency.

Most people know that what makes sounds different is that they have different frequencies.  Moreover, a high pitched sound has a higher frequency than a low pitched one.  Likely, that’s whence the idea of “high note” vs “low note” originates.

Every wave has not only a frequency, but also a wavelength.  The two go hand in hand according to the formula

    \[v=f\lambda\]

where
v means velocity (which, for this purpose, can be reduced to speed),
f means frequency
λ means wavelength

Dividing both sides by f, we arrive at a formula for the wavelength, \lambda:

    \[\frac{v}{f}=\lambda\]

As I mentioned in my January 13 post, the speed of sound s is given by

    \[s=(332 + 0.6T)m/s\]

where T is the temperature in degrees Celsius.

Let’s imagine, for temperature, a pleasant 20C. We have, for the speed of sound,

    \[s=(332+0.6(20))m/s=344m/s\]

Since speed and velocity are, for this purpose, the same, we can say that v=344.

Next, we need f, the frequency. Specifically, in this case, we want the frequency of “middle” F, which is meant to be the F just three keys above middle C. The physics dept at Michigan Tech provides this handy page, informing us that the F above middle C has frequency 349.23Hz.

With v=344 and f=349.23, we are ready to find the wavelength of middle F:

    \[\frac{v}{f}=\frac{344}{349.23}=0.985m=\lambda\]

Therefore, the wavelength of middle F, at 20°C, is 0.985m or 98.5cm.

HTH:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Perl programming: chop and chomp

Continuing with Perl programming, the tutor discusses a couple of simple yet important functions: chop and chomp.

Back in my Nov 19 post, I mentioned the Perl function chop. In that case, chop was used to strip the newline off the end of a name the user had inputted from the keyboard.

The chop function will “chop” off the last character of a string. Here’s an example:

#!/usr/bin/perl
$stringvar=”You look pretty:)”;
print $stringvar;
print “\n\n”; #Prints two empty lines for easier reading.
chop $stringvar;
print $stringvar;

should give the output

You look pretty:)

You look pretty:

Notice that, with no newline at the end of the string, chop removes the mouth of the smiley. The chop function always removes the last character, whether you want it gone or not.

On the other hand, the chomp function (apparently) removes only unseen characters. Therefore, it will remove a newline from the end of a string, but not a letter, number, or punctuation. Observe:

#!/usr/bin/perl
$newstringvar=”I love your purse:) \n”;
$question=”Where did you get it?”;
print $newstringvar;
print $question;
chomp $newstringvar;
chomp $question;
print “\n\n”;#Prints two empty lines for easier reading.
print $newstringvar;
print $question;

should hopefully produce the output

I love your purse:)
Where did you get it?

I love your purse:) Where did you get it?

The chomp function benignly left the second string alone, since its end character is a question mark. From the first string chomp did remove the newline character (denoted by \n).

Believe it or not, there is even more about chop and chomp. Therefore, I’ll be saying more about them in future posts:)

Source:

McGrath, Mike. Perl in easy steps. Southam: Computer Step, 2004.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Seasons: one Canadian’s definition of the first day of spring

The tutor has long been a student of the weather.  He shares his own definition and reflections about the coming of spring.

People who read this blog commonly may realize that I live on Canada’s west coast, but haven’t always.  The same is true for many you find out here.

Many of us likely don’t live here for the money.  Rather, the unique lifestyle attracts people – notably, the absence of the typical Canadian winter.

This year, I think I’ve had my snow shovel out twice, each time to push one or two inches of snow off the driveway.  It was as much for the mail carrier as for the car.  That was early in the season; the month of January, I can’t recall shovelling at all.

When I was a kid, I lived in the Maritimes.  Seasons change slowly there; winter comes late, but so does spring.  As a kid I realized that March 21 is only technically when spring arrives in the Maritimes.

At age 16 I arrived on the west coast in December.  The roads were bare; people didn’t wear winter boots or gloves.  Sometimes rain would fall many consecutive days, which surprised me.  However, when a sunny day came, you could play tennis or baseball just as if it was springtime.  Winter was similar to the other seasons: the weather on a given day affected what you could do, but the season itself didn’t.

That February I walked home from school in what a Maritimer would call “springtime” conditions.    Flowers were pushing up from their beds.  The evening air was delicious. I’d been told to expect it, but I wasn’t prepared.  Spring, officially, was still over a month away. To a Canadian from anywhere else, flowers in mid February seemed impossible.

Years went by.  I got used to the west coast weather.  I had kids.  I remember in the cold, raw wind and rain of a January morning, walking my three-year-old home from preschool.  He was crying; the 15 minute walk was against the wind, and it hurt his face. I knelt down beside him, pointing around the windswept field.  “Believe it or not,” I told him, “in six weeks, you’ll want to stay here and play.  It’ll be sunny and warm; you won’t want to go straight home.”  He didn’t believe me; I didn’t blame him.  We hurried through the soggy field and the January wind the last few minutes home.

Six weeks later, on the same walk home, my son was handing me his coat.  The green field was shining, radiating the bright sun’s heat.  The temperature was maybe eight or ten degrees Celsius.  Yet, it was a world away from the three degrees Celsius, with biting wind, we’d faced six weeks earlier.  Watching my son run around, I marveled at the arrival of spring.  Now, that raw weather of early January was hard to believe.

The Maritimer knows that, theoretically, spring starts on March 21, but it doesn’t necessarily translate to reality.  The west coast Canadian knows that, by March 21, it’s already been spring for around a month.  Clearly, a pragmatist might reject the March 21 definition as the start of spring in favour of a more meaningful one.

After 45 years shared between Canada’s east and west coasts, I’ve come up with my own definition for the first day of spring:  to me, it’s the first day that’s sunny with (positive) double digit temperature (Celsius).

This year, where I live, yesterday, January 25, was the first day of spring.  In the afternoon there was sun and blue sky with a temperature of 11C.  In fact, according to Environment Canada, yesterday’s temperature was 10C or 11C from around 10am to 4pm. The fields across the street have yet to turn firm and bright green, but they will in due time.

Why our weather on the west coast is so much milder than in the Maritimes, even though we’re further north, is bound to be an intriguing question to many Canadians. I’ll be exploring it in a future post:)

To my fellow Canadians: hang in there! Spring will come:)

Source:

Editor: Qentin H. Stanford. Canadian Oxford School Atlas, 6th ed. Toronto: Oxford    University Press.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Lifestyle: Reading tablets vs books

The tutor has mulled this issue for awhile.  Has the time come to discuss it?

More and more I’m hearing about reading tablets – tablets that display a book a page at a time.  When you’ve finished that page, you do some action that “flips” to the next page.  Having seen one, I seem to remember that the page content doesn’t just change to the new words.  Rather, there actually is some graphical effect that, for a split second, mimics a page flip.  Then, you’re on the next page.

Often, you might expect the younger generation to embrace a new gadget first. Interestingly, in my experience, the main proponents of these tablets are a generation older than I am.

The fans of the reading tablet make some compelling points.  You can carry hundreds of books in one tablet.  Since it’s always backlit, you can read from it even in the dark. There are numerous other advantages.

I confess I’m not tempted to buy a reading tablet.  From a logical point of view, perhaps I should be.

I haven’t read fiction for many years.  I do read books about computer programming.  I also do lots of “looking up” concepts or definitions from books.  Of course, some topics I look up on the internet.

Personally, I prefer to read from paper rather than a screen.  I like broad, soft cover books that take the shape of my lap when I’m lying back on the couch.  (I think) a book decorates a table.  If you are involved in several of them, you can leave them separately in various rooms, so there is one at hand whichever room you happen to be having your coffee.

Books take up space, of course, whereas e-books don’t.  With the current trend of miniaturization, every human being receives the opportunity to live in less space. Given how many human beings share the planet, the trend makes sense.

Yet, there are huge, open spaces where virtually no one wants to be, just as there are shelves of textbooks that few people want to read.  The dusty old academic loves to be just there, in that deserted room, surrounded by shelves of abandoned textbooks, inhaling that scent of paper.

I guess this blog will never be in print:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Statistics: how to enter a list, then find its mean, standard deviation, and median on the TI-83 Plus

Tutoring high school statistics, some of my students use the TI-83 Plus.  The tutor shows one way to find the mean, median, and standard deviation on it.

Of course, to find the mean, median, or standard deviation of a list of numbers on a calculator, you first have to enter them. As far as I can tell, the TI-83 Plus insists that you do so in a list. Therefore, it really helps to be comfortable with list creation and the list menus if you want to get the mean, median, and standard deviation on the TI-83 Plus.

Example:  Find the mean, median and standard deviation of the list of numbers
{-10.2, 55, 67, 100.9, 36, 120, -66}

Solution:  There are several ways to do this.  We’ll focus on one way in this post.

Step 1:  In between curly brackets, which you access by 2nd ( or 2nd ), enter the numbers, separated by commas. The comma key is just left of the left bracket key. Key in

2nd ( -10.2, 55, 67, 100.9, 36, 120, -66 2nd )

Step 2: Press the STO> button.

Step 3: Think of a name for your list. I called mine SUSAN. To key in the letters, first press 2nd, then the green ALPHA key. This will lock the keys in ALPHA mode. To name mine SUSAN, I keyed in the following:

2nd ALPHA LN 5 LN MATH LOG. You’ll see the letters in green above those keys.

Step 4: Press ENTER.

On my calculator, after I’ve successfully entered my list, it reprints across the screen. By pressing the blue sideways arrow keys (top right of the keypad), I can scroll through the list to check the values.

Assuming you’ve successfully entered the list, the hard part’s done:)

Step 5: To find the mean of the list, press 2nd STAT, then arrow across the top to the MATH choice.

Step 6: Press 3, which is the mean( option.

Step 7: Press 2nd STAT, then scroll down until you find your list name. Press ENTER to select it.

Step 8: You’ll find you’re back at the main screen. In my case, it says mean(lSUSAN. Close the bracket, then press ENTER.

Hopefully, you receive the answer 43.242857…

Step 9: To find the median, press 2nd STAT to return to the LIST menu. Once again, arrow over to MATH. You’ll see median( as choice 4. Key in 4.

Step 10: Retrieve your list by keying in 2nd STAT, then scrolling down to your list name and pressing ENTER to choose it.

Step 11: Back at the main screen again, I see median(lSUSAN. Close the bracket and press ENTER.

Hopefully, you receive the answer 55.

To find the standard deviation, key in 2nd STAT, then arrow over to MATH. You’ll see stdDev( is choice 7. Key in 7 to select it, then key in 2nd STAT and scroll down to your list. Press ENTER to choose your list. Back at the main screen, close the bracket and press ENTER.

Hopefully, you receive the answer 64.2407….

There are other ways to accomplish the same results on the TI-83 Plus. For more about them, and much more about using the TI-83 Plus in general, look to future posts:)

Source: TI-83 Plus GUIDEBOOK, Texas Instruments, 1999.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: fractions on the TI-83 Plus

The tutor first encountered the TI-83 Plus back in the 90s.  Tutoring math, some of my students are allowed to use it.

The TI-83 Plus is, to my mind, more like a computer than a calculator.  Most of my students aren’t allowed to use it on tests, yet some are.

One difference between a typcial scientific calculator and the TI-83 Plus is that, with the TI-83 Plus, so many of its useful functions need to be called from menus.  If you don’t know about a given menu, you may not have access to a function you need.  Such things are easy to find when you’re leisurely browsing from button to button with a cup of coffee, but often much harder to find when you’re unfamiliar with the calculator and writing a test.

An example is the fraction function, >Frac, which is located in the MATH menu. From the home screen (just press 2nd MODE to get there, then CLEAR if you need to), press the MATH button. You’ll see >Frac at the top: it’s choice 1. Now, press 2nd MODE to return to the main screen.

>Frac will normally be entered after a number or calculation.

Example: Find the answer to 2/3 + 1/11 – 3/4 in fraction form.

Solution:

Key in the following:

2÷3 + 1÷11 – 3÷4 MATH 1 ENTER

Hopefully you receive the answer 1/132.

You can also use >Frac to reduce a fraction:

Example: Give 289/561 in reduced form.

Solution: Key in 289÷561 MATH 1 ENTER

Hopefully, you receive the answer 17/33

You can even use >Frac to convert a decimal into a fraction.

Example: Give 0.428571428571….. as a fraction.

Solution: I find you need to give it around twelve or thirteen digits to convey the idea it’s a repeating decimal. Key in

0.428571428571 MATH 1 ENTER

Hopefully, you receive the answer 3/7

With such a plethora of useful functions, plus its graphing capabilities, the TI-83 Plus is almost too powerful for high school use. However, it will provide a veritable playground for future posts:)

Source: TI-83 Plus GUIDEBOOK, Texas Instruments, Inc., 1999.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Calculus: implicit differentiation

Tutoring calculus, this topic is of importance.  The tutor is happy to introduce implicit differentiation.

Implicit differentiation might come up a few weeks into the semester.  It’s a nice technique that enables the student to take derivatives of functions not solved for y.

Example:

Find the derivative of xy^2 -siny = 11

Solution: With implicit differentiation, we first assume that y is some function of x which we don’t know. We might imagine y=f(x). What we are trying to find is y', which might also be referred to as f'(x).

Following the point of view that y=f(x), we can rewrite the equation with f(x) instead of y:

    \[x(f(x))^2 - sin(f(x)) = 11\]

Now, we take the derivative from left to right on each side. x(f(x))^2 requires the product rule (uv)' = u'v + uv'

    \[(x(f(x))^2)'= 1(f(x))^2 + x(2f(x)\times f'(x))\]

Notice the chain rule in the second part: (f(x)^2)'=2f(x)\times f'(x). First, we take the derivative of the outer function with the power rule. Then, we multiply it by the derivative of f(x) itself, which is f'(x).

Next we take the derivative of -sin(f(x)), once again invoking the chain rule:

    \[(-sin(f(x)))'=-cos(f(x))\times f'(x)\]

On the right side, the derivative of 11 is 0. Writing the derivative of each term in the equation, we get

    \[(f(x))^2 + 2xf(x)\times f'(x) - cos(f(x))\times f'(x) = 0\]

What we are really trying to find is f'(x). We need to isolate it using algebra. First, we get all the terms that don’t include f'(x) onto the other side:

    \[2xf(x)\times f'(x) - cos(f(x))\times f'(x) = -(f(x))^2\]

Next, we factor out f'(x) from the left as a common factor:

    \[f'(x)[2xf(x) - cos(f(x))] = -(f(x))^2\]

Finally, we divide both sides by 2xf(x) - cos(f(x)) to isolate f'(x):

    \[f'(x)=\frac{-(f(x))^2}{2xf(x) - cos(f(x))}\]

Since the original assumption was that y=f(x), it follows that f'(x) is the derivative. If desired, we can substitute y and y' back into the solution so it matches the original context:

    \[y'=\frac{-y^2}{2xy-cosy}\]

HTH:)

Source:

Larson and Hostetler. Calculus: Part One. Toronto: D. C. Heath and Company, 1989.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.