English: in conflict, or in concert?

Tutoring English, you constantly need ideas on how to increase sentence variety. The English tutor discusses another possible variation….

In past English posts, I’ve referred to subordinating conjunctions, the complex sentence, and various other points of interest for those wanting to please the markers of their papers. Trying to satisfy an English teacher or professor might be an ongoing challenge.

Consider the following sentence:

In spite of the rain, they played the game.

It’s a complex sentence, of course, with a nice subordinate lead-in. In spite of prepares the reader for the surprise – that the game was played in the rain. The premise of the rain opposes the playing of the game; ironically, both happened anyway.

Could we pose those two apparently conflicting premises as actually supporting each other? Then, the surprise would be a different kind; the reader would realize that actually, the rain wasn’t seen as being an opposing factor:

With the rain falling, they played the game.

The word with suggests being together. In consequence, the feeling of the sentence is different. Perhaps the rain and the game were meant to proceed side-by-side.

Consider another example:

Although the children were playing loudly, the mother worked on her essay for night school.

As you’d probably expect, the children’s loud playing seems to conflict against the mother’s pursuit of her homework. What if, on the other hand, the two premises were posed as if they were expected to happen together?

Alongside the children’s loud playing, the mother worked on her essay for night school.

The sentence, rewritten thus, suggests that actually, the playing and the essay writing happened in concert.

The point is that two premises might seem naturally in conflict. However, they needn’t be posed that way. The surprising twist this approach can create will probably add spice to your writing if it’s not overused:)

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

Calculator features: built-in constants on the SHARP EL-520W

Tutoring physics or chemistry, you remember your attachment to your calculator as you pursued those subjects.  The tutor observes a feature of great convenience on the SHARP EL-520W.

In high school physics, you might be posed the following question:

\mbox{How many moles of electrons in a Coulomb?}

If you’re familiar with the concepts involved, the answer is not difficult.

    \[(\frac{6.24e^{18}\ electrons}{Coulomb})(\frac{1\ mole}{6.02e^{23}\ electrons})=\frac{6.24e^{18}\ mole}{6.02e^{23}\ Coulomb}\]

    \[=1.04e^{-5}\ moles/Coulomb\]

However, to approach the problem, you need to know the constants 6.24e^{18} and 6.02e^{23}, or else have them handy.

As a student, I soon memorized those constants. Before I had, I needed a reference table, which is often found in the cover of a textbook.

Now, you can use a calculator that has built-in constants. Today I’m highlighting the SHARP EL-520W. To get Avogadro’s constant – which is 6.02e^{23}, used in the above equation – simply press CNST, then enter 28. You’ll see 6.02...\mbox{x}10^{23} across the screen. To get 6.24e^{18}, the number of electrons in a Coulomb, you first need to realize that it’s the reciprocal of the elementary charge number, 1.60e^{-19}. To get the elementary charge number, press CNST, then 09. You’ll see 1.60...\mbox{x}10^{-19} across the screen.

The SHARP EL-520W has 52 built-in constants, many of which I’ve never had occasion to use. However, I’ll discuss more about them in future posts:)

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

Organic Chemistry: naming alcohols

At university, the tutor encountered the world of organic chemistry – as many of you have or will.  Although it comes up rarely in high school tutoring, a basis is beneficial before you face it at post-secondary level.

If you haven’t read my earlier organic chemistry articles here and here, you may wish to do so. Continuing from them, we look today at naming alcohols.

Example 1: Name this molecule.

Step 1: Count the carbons in the chain. In this case, we have 9.

Step 2: Refer to the table in this post, which gives the name of the chain based on its number of carbons.

Step 3: Counting from the end of the chain closest to the OH, decide which carbon the OH is on. In this case, OH is on the fourth carbon.

Step 4: The name starts with the number of the carbon that has the OH, followed by a hyphen, then the chain name from Step 1. At the end of the chain, replace -ane with -ol, which means the molecule is an alcohol.

In our example, the OH is on the fourth carbon. The chain has nine carbons, suggesting nonane. Therefore, our molecule is 4-nonanol.

OH means alcohol in the context of organic chemistry. Not all molecules with OH are alcohols; nevertheless, OH is called “the alcohol group”.

Here are a couple more alcohols:

A final example reminds the student to count from the end of the chain that is nearest the alcohol group:

Many people might mistake the alcohol above for 5-hexanol. However, since the OH is nearer the bottom, you count up from there.

There are many more complicated alcohols – and of course other organic molecules – to name. Look for them in future posts:)

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

Math: simplifying cube roots

Tutoring high school math, cube roots come up a few times a year. The math tutor opens the discussion on simplifying them.

By way of introduction, you might want to read my short article here on perfect cubes and cube roots.

Of course, \sqrt[3]{8}=2 because (2)(2)(2)=2^3=8. Note that 2^3 is said as “two cubed”.

What about \sqrt[3]{40}? There is no whole number x such that x^3=40. Yet, \sqrt[3]{40} can be simplified.

Step 1: Find the factor of 40 that you can take the cube root of (8 in this case).

Step 2: Rewrite \sqrt[3]{40} as \sqrt[3]{8}\sqrt[3]{5}.

Step 3: Replace \sqrt[3]{8} with 2.

We arrive at

(1)   \begin{equation*}\sqrt[3]{40}=\sqrt[3]{8}\sqrt[3]{5}=2\sqrt[3]{5}\end{equation*}

The trick becomes identifying which perfect cube can be factored from the number.

Example: Simplify \sqrt[3]{192}

Solution:

In this case, realize that 192=(64)(3). Therefore,

(2)   \begin{equation*}\sqrt[3]{192}=\sqrt[3]{64}\sqrt[3]{3}\end{equation*}

Now, \sqrt[3]{64}=4. Therefore,

(3)   \begin{equation*}\sqrt[3]{64}\sqrt[3]{3}=4\sqrt[3]{3}\end{equation*}

How we discover 64 as the number to use, and more, will be explored in a post coming soon:)

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

Math: multiplying with algebra tiles

The tutor continues with algebra tiles.  Tutoring math, you see them especially in grades 9 or 10.

I began discussing algebra tiles here. Recall that they are meant to model numbers, variables, and operations you can perform to them. Recalling the foil method, we’ll take a look at how algebra tiles can be used to “show” it.

Example: Show (x+6)(x-2) with algebra tiles.

Solution: We know by the foil method that the answer is x^2+4x-12.

With algebra tiles, the layout is rows and columns. We might, for instance, display x+6 across the top, and x-2 down the left, as follows:

The result of the multiplication is in the “body” of the table. We see one large white square (which means +x^2), six white rectangles (+6x), then two black rectangles (-2x), and finally, twelve small black squares (-12). The two black rectangles cancel out two of the white ones, leaving four. The result is x^2 +4x -12, as we knew from foil.

Some people find foil quite difficult, while this visual method is easier for them. Once they’re used to it, they hopefully become comfortable with how it translates to foil.

I’ll be saying more about algebra tiles in future posts:)

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

Math: adding and subtracting radicals

Tutoring high school math, radicals are prominent.  The math tutor introduces addition and subtraction of them.

Adding radicals is much like adding variables. Note that x=1x; the 1 is understood to be there, but never written. Similarly, \sqrt{3}=1\sqrt{3}. Therefore, 2x +x=3x; similarly, 2\sqrt{5} + \sqrt{5}=3\sqrt{5}.

You can’t simplify x plus y; x+y is just x+y. Similarly, \sqrt{6}+\sqrt{7} doesn’t simplify; it’s just \sqrt{6}+\sqrt{7}. A calculator will give you a decimal for it, but that’s not an exact value.

So what about the following:

Example 1: Simplify \sqrt{12}+\sqrt{75}+\sqrt{3}

You may want to read up on simplifying radicals in my earlier post to understand what follows.

Solution: On the face of it, we can’t add the radicals because they’re not the same kind. However, we can simplify some of them as follows:

(1)   \begin{equation*}\sqrt{12}=\sqrt{4}\sqrt{3}=2\sqrt{3}\end{equation*}

(2)   \begin{equation*}\sqrt{75}=\sqrt{25}\sqrt{3}=5\sqrt{3}\end{equation*}

Simplifying reveals them to be the same kind. We indeed can add them, as follows:

(3)   \begin{equation*}$\sqrt{12}+\sqrt{75}+\sqrt{3}=2\sqrt{3}+5\sqrt{3}+\sqrt{3}=8\sqrt{3}\end{equation*}

By a similar principle, \sqrt{50}-\sqrt{8}=5\sqrt{2}-2\sqrt{2}=3\sqrt{2}.

I’ll be covering some different examples in an upcoming post:)

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

Organic Chemistry: Simplified drawings

When you tutor high school chemistry, you cover a little organic chemistry. Today we visit a topic that rarely gets asked about during tutoring, but is very useful to know at the college level.

 
In my previous post on organic chemistry, I introduced the topic and showed a few drawings. They were of this style:

Recall that carbon, which appears as C in the diagrams, makes four bonds. (Each bond is a line to another atom.) Notice that most of the bonds are between carbon and hydrogen (H).

Since bonds usually are between C and H, chemists often don’t draw the H atoms. They’re understood to be there unless a bond clearly goes to something else. In such cases, only the bonds between carbons or between carbons and other atoms are shown, like so:

With the hydrogens left out, the diagram becomes simpler to understand. For example, you can more easily tell the how many carbons it has.

Some people take simplification a step further, writing only sticks for the carbon skeleton. Each corner represents a carbon atom. Therefore, all three of the following diagrams show butane:

The simplified drawing styles really come in handy for showing molecules with features such as alcohol. Here’s an example:

How we know the molecule is 2-pentanol will be covered in a future post:)

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

Math: algebra tiles

Tutoring math, you might explain algebra tiles a few times a year.  The math tutor introduces them.

 
If algebra tiles existed when I was in school, I never saw them. Everything back then was symbolic, and teaching didn’t have the directive of making the learning easier.

Nowadays, attitudes towards teaching and learning couldn’t be more different. Manipulatives, as they are called, are used to explain counting and even variables in a visual, physical way. An abacus could be used thus.

Algebra tiles are used to depict expressions with variables. A big square means x^2, a rectangle means x, and a small square means 1. Consider the following example:

Example 1: Show 3x^2 + 4x +5 with algebra tiles.

Solution: We need three big squares for the 3x^2, then four rectangles to make 4x, and finally five small squares to indicate 5. Like so:

Depending on the situation, you may need to show the rectangles horizontally and/or group the small squares in various different ways. Here is another depiction of 3x^2 + 4x +5:

How do you show negative values? With color, is the answer. Often, the light coloured tiles are taken to mean positive, while the darker ones mean negative.

Example 2: Show -2x^2 -x + 3 with algebra tiles.

That’s a pretty good beginning with algebra tiles. In future posts I’ll show some of the math concepts you can illustrate with them:)

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

English: subordinating conjunctions: while

Tutoring English, a constant question is how to keep writing fresh.  The English tutor discusses his recent satisfaction with the word while….

Continuing, generally, the discussion about complex sentences that began with my last post, I’d like to tell you about my rediscovery of the word while.

Last summer, I had several writing projects to complete. Needing, as always, to maintain sentence variety, yet without unnecessary length, I gradually zeroed in on while. I never used it much in high school or university, but now I’m finding new uses for it weekly.

While is a great word: I think the reason is that it has two potential meanings. It can mean “at the same time” or “although.” Often, those two meanings apply simultaneously.

Consider the following example:

While he hated math, his highest mark was in it.

For a situation to be ironic, two conflicting premises need to be true at the same time. That’s why while, I’m finding, shines among subordinating conjunctions.

While India’s economic growth rate may lag China’s some years, India is ahead in being a democracy.

In one of my writing endeavours last summer, the target was really stingy with word allowance. Being a single word with two potential meanings, while came to my rescue time and again. Maybe it will for you, too:)

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

English: the complex sentence

Tutoring high school English, sentence variety is a constant concern.  It’s the same situation for college students and writers.  The English tutor shares one commonly known – and appreciated – type of sentence.

Definitely, one preferred structure in high school English is what I’d call the complex sentence.  It has two parts: one dependent, one independent.  (The independent part can stand alone as a sentence, while the dependent part cannot.)  Let’s look at an example:

Because she set the burner too high, the eggs are rubbery.

In the sentence above, Because she set the burner too high is the dependent part (aka, dependent clause). It’s clearly a reason, but doesn’t make sense unless you know what for. The independent part (clause) is the eggs are rubbery . “The eggs are rubbery” has a subject and an action; it’s a standalone sentence. It’s not dependent on another idea because it doesn’t have a subordinating conjunction.

The subordinating conjunction in the sentence above is because. It makes she set the burner too high a reason for the other premise – that the eggs are rubbery. In a complex sentence, the subordinate – aka, dependent – clause exists to somehow serve the independent one. By “serve”, we mean give it more meaning. The subordinate clause in the example above explains the reason behind the independent one. However, the subordinate clause can relate to the independent one in other ways as well. For instance, the subordinate clause often expresses surprise or irony, as in the next example:

The ball game played as scheduled, although it was a rainy evening.

Once again, the dependent – or subordinate – clause is typed in green. In this example, the subordinating conjunction is although. It expresses surprise that, even with the rain, the game went on.

There are many common subordinating conjunctions: after, although, because, before, once, since, unless, while…., just to name a few.

When written correctly, complex sentences are generally appreciated by teachers as showing an advanced level of writing. Therefore, they’re worth understanding and getting right.

Now that I’ve started this thread, I’ll be talking more about it soon. Until then, enjoy this spectacular spring day!

Source: Hodges, Webb, Miller, Stubbs: Harbrace Handbook for Canadians.                              Scarborough: Nelson Education Ltd., 2003.

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