Math: memory calculations on the Sharp EL-520W

Tutoring math, you want to optimize your students’ calculator usage.  The tutor recalls his own use of the calculator memory back in high school; it was indispensable then.

Let’s imagine you’re facing the following situation:

(3√(25+√14)^2 + (sin(0.27) +2)^3 – 3√(1/7 – √(17))2

True, you could just enter it, start to finish, into a forward entry calculator like the Sharp EL-520W. Assuming you enter everything perfectly, you’ll get the right answer: the scientific calculator will do the order of operations correctly.

With such a long and complex expression, however, the probability of entering something wrong is typically high. One wrong keystroke will result in the wrong answer. What a shame, following so much effort spent trying to enter the expression correctly!

The tutor admits that he’s as likely as anyone to make an entry error. How, then, do I increase the reliability of my entries?

For a long calculation like above, one strategy is to break it into parts, get a reliable answer for each part, then add or subtract them to arrive at the final answer. Let’s revisit our expression from above.

(3√(25)+√(14))^2 + (sin(0.27) +2)^3 – 3√(1/7 – √(17))2

One approach is to get the answer to (3√(25)+√(14))^2. This expression, by itself, is short enough to enter reliably. Even so, I enter it a few times, until I get the same answer over again:

(3√(25)+√(14))2=44.43122487…

I do the same with the second part:

(sin(0.27) +2)^3=11.6466….(calculator in rads)

Similarly with the third part:

3√(1/7 – √(17))^2=2.5115….

Now, I add (or subtract) the three partial answers together to get the final one:

44.4312….+11.6466….-2.5115….=53.56631271

At this point, some of my students will put the brakes on: “My teacher says I’m not allowed to round until the end. How do I get an exact answer if I have to do the computation in stages?”

While you could write each partial answer down from the calculator screen, keeping all ten digits each time, then re-enter them to get the final answer, I recognize that’s not practical. Worse yet, it’s downright error-prone: you could (and I likely would) make an error copying the answers down.

What you can do, instead, is use the calculator memory to store the partial answers, then recall them as needed to get the final answer. The key sequence to store an answer from the screen is STO letterkey. For instance, to store √(18) in letter E, you’d key in square root 18, then the equals key, then STO, then finally the “log” key (just above the “log” key you’ll see a green ‘E’). To recall that answer later, key in RCL then the “log” key again.

To apply this memory feature to our procedure above: get the answer to (3√(25)+√(14))2, then key in STO CNST to store the answer in the ‘A’ variable. Continue, storing the next partial answer in ‘B’, the last one in C. Now, to get the final answer, key in RCL CNST + RCL y^x (for ‘B’) – RCL x^2 (for ‘C’). You, too, should arrive at 53.56631271.

You can store on top of old values as new calculations demand.

The memory feature can make long calculations much easier and safer. Enjoy playing with it:)

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

Financial math: mortgage payments

The tutor notices that financial math is more prominent in high school than it was twenty years ago.   Probably the change is good; let’s embrace it with a first look at mortgage payments. This topic comes up annually in tutoring.

Even before the advent of financial calculators, people had to calculate annuity payments, mortgage payments, etc.  Such financial products have existed certainly since the industrial age – probably even before.

The formula to calculate a mortgage payment isn’t that difficult to use, but probably outside the comfort zone of most consumers.  The obvious question:  How, in the 70s, did people calculate mortgage payments, when financial calculators – if they existed at all – were rare?

The simple solution:  they used mortgage tables.  You looked up the interest rate and the term (in years) of repayment.  The amount you’d arrive at would be the payment per thousand dollars of the loan.  They were called mortgage tables, but you could use them for any loan.

I’ve found a table I like the look of over at realsavvyrealestate.com. Let’s put it to work:

Example: Calculate the mortgage payment for this case:

loan 300 000
interest 6.5%
term 25 years
payment frequency monthly

On the table from realsavvyrealestate.com., we look down the left column to find 6.5%. From there, we look across to the 25 year column. The number we arrive at is 6.75207, which means that the payment is $6.75207 per thousand dollars of loan. The loan amount in this case is $300 000. Therefore, the monthly payment will be

$6.75207×300=$2025.62

Mortgage tables enable easy loan payment calculations. Many people from an earlier generation – myself included – can’t help but prefer them even now.

Mortgage tables, of course, can’t handle the diversity of financial situations that a dedicated calculator can. In coming posts I’ll explore the use of a financial calculator – and perhaps even the option of doing the calculations using financial formulas.

Cheers:)

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

Math: a sales tax problem feat. the “let statement”

Tutoring math, you know sales tax problems to be part of the elementary school repertoire.  The tutor brings forward a more subtle one that’s probably grade 10 or 11 level.

Sales tax problems were calculated by hand when I was in elementary school.  Two cultural changes have occurred since then:

1)  The widespread use of calculators, often even during elementary years.

2)  The use of cards instead of cash.

Both changes have diluted people’s consciousness of sales tax – but especially, I think, the second one.

In any case, sales tax word problems still occur now and then.  Here’s one that’s a little different from merely calculating the tax and adding it to the sticker price:

Example: If the total price paid, including tax, is a hundred dollars, what must the sticker price have been?

Solution: The first step is to invoke the all-powerful “let statement”:

Let x=price before tax

The tax, assuming 12 percent sales tax, will be 0.12x. Therefore,

Total=price + tax

becomes, more specifically,

100=x+0.12x

Since x=1x, this simplifies to

100=1.12x

We now divide both sides by 1.12:

100/1.12 = x

leading to

89.29=x

So, the sticker price must have been $89.29 to give the final price, with tax, of $100.

The “let statement” is what makes this approach straightforward. It gives direction to the solver, and clarity to both solver and marker. When facing word problems, the importance of the “let statement” cannot be overemphasized:)

Hope you had a great Thanksgiving. Cheers!

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

Chemistry: how soap works

The tutor remembers, in a lab, receiving this basic explanation from a grad student.  While I’ve never fielded this question during tutoring, it might be of general interest.

 
A primary fact of chemistry is that “like dissolves like”. Yet, soap doesn’t dissolve dirt; it emuslfies it. So we’ll alter the principle to “like attracts like”.

A reader might protest; don’t opposites attract? Opposite charges do, yes, but opposite bond types do not.

The bond type of water is essentially polar. The oxygen hoards the electrons, leaving the hydrogen nuclei relatively bare. Therefore, the oxygen end of a water molecule is relatively negative, while the hydrogen end is correspondingly positive.

In contrast to water, the bond type in oils is non-polar. The electrons remain more or less even throughout, preventing localized charges from developing. Any cook knows that “oil and water don’t mix.” Now you know why: water is polar, while oil is non-polar. From a molecular point of view, like attracts like; opposites repel.

You know water alone won’t clean dirty hands, dirty clothes, or dirty dishes. That’s because the fundamental component of “dirt” is oil (grease being an example). Soap is a chemical “middle man” that is attracted both to water and to oil; hence, soapy water will clean off the dirt.

How can soap attract both water and oil, which are basically opposites? Soap begins as an oil, but is chemically treated (often with sodium hydroxide) so that one of its “ends” receives a polar bond. The end with the polar bond attracts water.

At the same time, the other end of the soap molecule maintains its non-polar character. This non-polar end attracts “dirt” (oil, grease, earth, or what have you).

Soap has to be wet to work. In such a setting, the non-polar ends of the soap molecules latch onto the grease molecules. Then, the polar ends of the soap molecules, which are “attached” to water molecules, get pulled in different directions by the motion of the water. As the soap molecules are pulled away, they carry the grease molecules with them; the lumps of grease are thus pulled apart into smaller units. This process is called emulsification; as it repeats, the grease droplets become tiny enough to be completely removed by the water.

The classic “bubble” you get from soap is, essentially, a “water-free” zone surrounded by the nonpolar ends of soap molecules. Water, in turn, surrounds the bubble, since it’s attracted to the polar ends of those same soap molecules.

With potentially so many dishes to wash tonight (the Canadian Thanksgiving), this article might provide some “food for thought” for anyone peering into the bubbles:)

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

Weather and seasons: autumn feat. length of day

Seeing the mist in the trees, the tutor reflects on the arrival of autumn.  What’s the significance of autumn – and to whom is it yet important?  What’s the connection between autumn and the other seasons, and what’s the “other day” in the year that’s connected with this one?  While these questions don’t come up in tutoring, they are fundamental to any culture centred on nature.

As anyone reading my blog for some time might suspect, I lived in farming regions as a kid.  Farmers are not necessarily seen as nature-centred, but of course they are. The farmers I knew owned family operations that were partially – or mainly – subsistence.   Those farms were mixed:  they had some orchard, some potatoes, some carrots or beets, some corn, some hay, etc, along with pigs, cows, and chickens.  The richer farmers had dairy operations – but they, too, had large fields of other crops.

With so many crops to plant, take care of, and harvest, the farmers’ kids were constantly aware of the seasons.  At school, as well, the seasons were in constant focus.  I wasn’t a farmer, but it gets in your blood.  To this day, I’m mindful of the season and its significance.  Ironically, where I live now isn’t much oriented towards farming.

To begin with, what’s the significant difference between the spring and summer seasons, as opposed to the fall and winter ones?  The answer is length of day:  during the spring and summer, more than half the 24-hour day is between sunrise and sunset. During fall and winter, the time between sunrise and sunset is less than twelve hours.

During autumn, the days continually shorten – although the rate slackens as you approach the shortest day, which is the beginning of winter.  Then, the days start to lengthen, reaching 12 hours on the day spring begins.  (The timing may not be exact everywhere, but fundamentally it’s true.)

With that in mind, there must be a day in winter – near the end of it – that’s the same length as today.  Counting the days since fall began (on September 23 this year, if I recall), there’ve been 17, today included.  The equivalent day on the other side should be 17 days before spring starts.  This year, it starts on March 20th; therefore, March 3rd should be the about the same length (in daylight) as today.

The people at the National Research Council of Canada have a great gadget called a sunrise/sunset calculator. I checked it for today (October 10) and March 3.  Here are the results:

Day Sunrise to Sunset (hrs)
October 10 11.11
March 3 11.15

I think we’ll agree the day lengths match pretty closely:)

Autumn might be my favourite season. What academic can fail to like it, given its connection with school? Of course, Halloween – my favourite holiday – is in autumn. With the fresh, brisk air, the smell of wood smoke, the fallen leaves skidding along the pavement, the early darkness – the fall is a magical time.

Look for more on the weather and seasons in future posts.

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

Exercise: the cool-down: part one

The tutor realizes that not everyone is interested in computer science.  Exercise is a constant topic of conversation today, if not the most pressing during tutoring….

Those who exercise vigorously are trained to “cool-down” afterwards. Typically, the cool-down is a period of light-to-moderate exercise of around 5 minutes. It’s done at a comfortable pace – maybe 20 to 40 percent of maximum. The point is to decrease the exertion, yet keep the circulation elevated. You might hear people take slow, deep breaths during the cool-down.

If the cool down is not done, the exerciser might experience empasized muscle stiffness and discomfort following the workout. What does the cool-down accomplish; why is it so important?

I’m not trained in the science of exercise, but I’m a veteran of it. Here’s some of what I’ve been able to piece together to explain the importance of cooling down:

Oxygen debt typically occurs during vigorous exercise, especially when bursts of power are demanded (as happens in soccer, hockey, football, weight training, sprinting, etc). In a period of oxygen debt, the muscles work faster than the heart and lungs can supply them with oxygen. If the muscles undergo oxygen debt, they only partially break down the glucose; lactic acid is left over.

During the cool-down, the exertion level is much decreased, once again giving the heart and lungs easy opportunity to supply enough oxygen to the working muscles. Therefore, the lactic acid buildup ceases. At the same time, the muscle cells flush the lactic acid into the bloodstream. The still-elevated level of circulation during the cool-down enables the blood to carry the lactic acid from muscles to the liver. At the liver, the lactic acid is actually converted back to glucose.

So it seems that the body uses the cool-down to shuttle the lactic acid from the muscles to the liver. This is one benefit of the cool-down, but it’s not the whole story. I’ll provide the next installment in a future post:)

Source: Wikipedia.

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

English: rhymes

Tutoring English, it’s fun to share little definitions that expand on topics you thought were familiar.  The tutor shares a few about rhymes.

Most people know what a rhyme is:

Don’t spread the lime
Until it’s time.

This simple example is a case of end rhyme.  Next, there’s internal rhyme:

She told the date and set us straight:
“Set clocks back on the Second.”#see below

So far, most people may not be surprised.  However, did you know there’s masculine rhyme?

We called, it came:
The bird was tame.

With the masculine rhyme, only one syllable (the very end one) rhymes, whereas in feminine rhyme, two syllables sound the same:

She didn’t fancy biking;
She did enjoy hiking.

There’s even beginning rhyme, with the syllables that agree at the front of each line:

Bread we have:
Fed they’ll be.

Believe it or not, there’s still more about rhymes.  I’ve got a couple more definitions to run down; the sources don’t agree on them (so far).  However, I’ll keep you posted:)

Source:  Literary Terms, Coles Notes.  Toronto:  Indigo Books and Music Inc, 2009.

#Source:timeanddate.com

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

Perl: the if, [elsif], else construction

Tutoring computer science, this topic is fundamental.  The tutor gives a short program as a focal point.

 
Back in my last post I introduced the if, else construction. Today we’ll look at a sample program that implements it.

Let’s imagine that your (geeky) spouse has written a grocery list into a Perl array. However, you don’t know if she’s included olive oil. You write a little program to read the array, then report if “olive oil” is present.

One extra parameter: your spouse likes to abbreviate. Therefore, she might have written “olive o.” for “olive oil.”

Here’s the program. Note that the comments are in green.

#!/usr/bin/perl

#for our purposes, you might imagine you can’t see the
#contents of @groceries

@groceries=(“butter”,”sugar”,”bread”,”eggs”,”deli ham”,”cheddar cheese”,”olive oil”,”coffee”,”oranges”,”apples”,”grapes”,”olive o.”,”milk”,”cream”);
$i=0;
while($groceries[$i]){
if($groceries[$i] eq “olive oil”){
print “Olive oil is element $i in the list.\n”;
}
elsif($groceries[$i] eq “olive o.”){
print “Olive o. is element $i in the list.\n”;
}
else{
print “Not element $i\n”;
}
$i++;
}

Let’s imagine this program is called groclist.txt. On the command line, in its proper directory, you’d run it by keying in

perl groclist.txt

Assuming no typos (by me or you), you might be greeted with the following output:


Not element 0
Not element 1
Not element 2
Not element 3
Not element 4
Not element 5
Olive oil is element 6 in the list.
Not element 7
Not element 8
Not element 9
Not element 10
Olive o. is element 11 in the list.
Not element 12
Not element 13


The details of this program I’ll explain next post. Cheers:)

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

Perl: $variable status: the “if” test

Tutoring computer programming, some issues that seem negligible become very important to consider.  The tutor discusses the status of variables.

 
In computer programming blogs, as well as in some books, you’ll often see the construction

if($variable)
{
these instructions;
}
else
{
these instructions;
}

The idea is that if $variable has been defined and given a non-zero value, for example

$variable=14;

or

$variable=”hello”;

then the program will execute the “if” set of instructions. However, if

$variable=0;

or $variable is simply not declared in the program, the “else” set of instructions will be executed.

As we’ll see in coming posts, this “if” test is very convenient. Moreover, you see it so often in articles about programming, you might think it’s generic. However, not all programming languages support the “if” test as written above. Rather, some require an expression in the parentheses, such as

if($variable equals value)

or

if($variable not equal to value)

Happily, Perl does support the if($variable) test. I wonder if some writers who use it in articles are Perl programmers who forget that it doesn’t work in some other languages.

I’ll be exploring the “if”, “else” construction, as well as the “if” test, in future posts. Cheers:)

Source: Robert’s Perl Tutorial

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

Perl programming: follow-up from Sept 29

Tutoring math or sciences, you need to remind your students to “show their work.”  The tutor observes the same precedent in computer science.

Back in my last article, I discussed the changes to the Perl compound interest calculator so it could cover all the compounding possibilities. Though it accomplished its objective, that discussion left some loose ends, one of which I’ll attend to now.

Issue 1: Notice the “#” sign in the sixth line of code:

$ppyear=$ARGV[3];#compounding periods per year

Normally, you can’t just add text to a computer instruction and have the program still work. However, in Perl, “#” means “comment”: following the “#” sign, the rest of the line will be ignored by the computer.

Comments are used to tell the reader the functionality of lines of code. In the example above, $ppyear=$ARGV[3]; introduces a new variable. The comment after the “#” tells its purpose.

Even with small programs, the functionality of certain lines may be hard to decipher at a glance – even for the author of the code! If a few weeks have passed since you’ve looked at your program, you might not easily recall what a given line does. Part of the skill of programming is knowing what lines need to be explained, as well as how to explain them concisely with little comments. Even one or two good comments can make a program much easier to understand at a glance.

Putting the discussion into scholastic perspective, comments are how programmers “show their work.” Showing one’s work is really about explaining the ideas behind what’s written down. When the ideas are known, marks can be easier to give – and programs can be easier to fix. When you’re unsure of what you’re doing – that can be the best time to offer explanation:)

Source: Robert Pepper’s Perl Tutorial.

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