Number Sets: Naturals, Wholes, Integers, etc

Tutoring math, you often get asked about sets of numbers.  Let’s sort out what belongs where.

We’ll make this story as short as possible:

Naturals (N):  {1,2,3,4…..}  These might be referred to as counting numbers.

Wholes (W):  {0,1,2,3,4…..}  These include all the naturals, plus zero.

Integers (Z):  {….-3,-2,-1,0,1,2,3….}  These include all the whole numbers, plus the negatives of them.

Rationals (Q):  You can’t list these numbers in order, since there is always another one between any two you name.  However, you can define them as follows:  a rational number consists of any integer divided by an integer other than zero.

In other words,

(1)   \begin{equation*}rational=\frac{integer1}{integer2}\end{equation*}

where integer1 can be zero, but integer2 cannot be zero.  Therefore, rationals include the following examples:

Hence, we see that any integer, since it can be written as itself over 1, is rational.

It turns out that rationals also include repeating decimals as well as terminating ones. You can verify the facts on your calculator:

(2)   \begin{equation*}\frac{409}{99}=4.1313131313.....\end{equation*}

and of course

(3)   \begin{equation*}\frac{-12}{5}=-2.4\end{equation*}

Up to and including the rationals, each set contains the previous one.  That is, the rationals contain the integers, the integers contain the wholes and the wholes contain the naturals.  However, the next set of numbers – called the Irrationals – is completely different from the rationals and separate from them.  The irrationals contain non-repeating, non-terminating decimals.  These numbers are written symbolically:  examples are √11, as well as our friend pi (∏).

The Real Numbers (R) contain all the rationals, plus all the irrationals.  There is yet another set:  the Imaginary Numbers.  We’ll save them for another post.

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

Multiplying two binomials: the FOIL method

Tutoring math, the FOIL method is used daily.  We’ll explain it here.

In math, the distributive property is used constantly.  An illustration:

(1)   \begin{equation*}3(5x-7)=15x-21\end{equation*}

To see what’s really happening here, look to the following:

We see that we take the outer number (in this case, 3) and multiply it by each term in the brackets, writing the result each time.

A binomial is a couple of terms that cannot be added. In the above example, 5x-7 is a binomial. When two binomials are multiplied together, it might look like this:

In order to multiply a binomial by a binomial, notice the steps below:

(The letters stand for First, Outer, Inner, Last: hence the acronym FOIL)

F: 2x times 4x gives 8x2
O: 2x times 9 gives 18x
I: -5 times 4x gives -20x
L: -5 times 9 gives -45

Writing the terms consecutively, we get

8x2 + 18x -20x -45

Now, we can combine the 18x with the -20x to finally obtain

8x2 -2x -45.

Indeed, (2x-5)(4x+9) = 8x2 -2x -45.

To verify the FOIL method, try it on (12)(10), written as

(10 +2)(9 + 1)

F: 10(9)=90
O: 10(1)=10
I: 2(9)=18
L: 2(1)=2

90 + 10 + 18 + 2 = 120, which of course we know is the answer to (12)(10).

Hope this helps.

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

Physics: Newton’s First Law

Tutoring high school physics, you get the privilege of retelling Newton’s three Laws.  They can’t be discussed too often.

Newton’s First Law:

If no unbalanced force acts on an object, it either continues moving in a straight line at constant speed, or else remains in its state of rest.

Like his other laws, Newton’s First Law contains some surprises if you examine it closely.  First of all, how could he have predicted it, when nothing on Earth does continue moving at a constant rate when left alone?

Back in high school, one of my textbooks (I can’t remember which one; it was over twenty years ago!) explained that Newton understood the tendency to retain constant velocity by comparing a ball to a brick.  He noticed that when you throw a brick, it stops almost immediately when it hits the ground.  However, a ball might continue moving for a long time after it contacts the ground.

Newton realized that the difference between the ball and the brick is that friction acts more emphatically on the brick.  Given its shape, the brick catches the edges of the ground’s surface.  The ground grips the brick, stopping it.  The ball, with its round shape, makes smooth contact with the ground rather than rubbing along it.  Friction is, of course, resistance to rubbing, so the brick’s movement across the ground is much more affected by friction than is the ball’s.

Newton realized that air resistance is just another case of friction.  Therefore, everything moving on Earth – including the ball – is eventually arrested by friction.  Friction is unbalanced unless you apply force to counteract it.  Hence, a cyclist can continue at a constant speed if he or she is willing to pedal against friction.  However, when that cyclist stops pedalling, the air gradually halts him/her.

In space, there is no air resistance, so objects can (and do) continue their straight line motion forever.  Gravity can make things change direction, but gravity is then an unbalanced force.  Unbalanced means that there is not an equivalent force opposing it.

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

Chemistry: Covalent Compounds

When you tutor high school chemistry, writing chemical formulas comes up every year.  Having investigated ionic compounds, let’s sort out covalent ones.

If you look in the Chemistry category, you’ll see various posts about writing chemical formulas for ionic compounds.  Those posts explain that, for ionic compounds, matching the combining capacities of the metals and the nonmetals is key to writing proper formulas.  Another way to look at it is that the charges must be balanced in a viable ionic compound formula.

With covalent compounds, we needn’t worry about combining capacities; the name of the compound tells all.  That’s because covalent names include prefixes that tell us the number of each type of atom involved:

prefix number
mono 1
di 2
tri 3
tetra 4
penta 5
hexa 6
hepta 7
octa 8
nona 9
deca 10

Take, for example, dinitrogen pentoxide. The di before nitrogen tells us there are two nitrogens; the penta before oxide tells us there are five oxygens.  Therefore, we have

dinitrogen pentoxide: N2O5

Going the other way, here is a point to mind: The prefix mono is rarely used. For example:

SF :  sulphur hexafluoride.

Note that we don’t call it monosulphur hexafluoride, even though there is only one sulphur. However, we do call CO carbon monoxide. To my knowledge, mono is only used when another related compound is more common. In the case of carbon monoxide, we use mono to distinguish it from carbon dioxide, which of course is more often mentioned. Whatever the reason, the prefix mono is seldom used, even when there is only one of that atom.  Other examples:

CCl4 : carbon tetrachloride

BF3 : boron trifluoride

Of course, one question that might evolve:  “How can I tell if a compound is ionic or covalent?”  The simple answer:  If the compound is ionic, its name starts with a metal.  If it’s covalent, it starts with a nonmetal.  I explain how to tell a metal from a nonmetal here.

Remember:  only covalent compounds use the numeric prefixes in the table above; ionic compounds don’t use them.

Source: Chemistry, Charles E. Mortimer, Sixth Edition, Wadsworth, Inc., 1986.

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

Cell Biology: Function of the Cell Membrane

When you tutor high school biology, cell biology is a mainstay.  The particular role of the cell membrane sometimes gets neglected.

Membranes are very important in the construction of organisms.  Every membrane has the basic function of keeping materials separate or else keeping them where they are.  Every cell has a membrane surrounding (enveloping) it.  Some people call it the cell membrane; others call it the plasma membrane.

The role of the cell membrane is sometimes described as protection – that it shelters the cell from its environment.  To some extent you could argue that’s true, but really the cell membrane’s function is to control the movement of materials in and out of the cell.

You can think of the cell as a factory.  It has valuable things inside that need to be there.  Some of those valuables are destined to leave, but only under the correct circumstances.  Until the appropriate time of departure, those valuables must remain safe inside.

The factory, however, can’t just be locked up until someone comes to take delivery of the merchandise.  New supplies are arriving all the time, and need to be taken in.  However, anything (and anyone) not meant to be there is stopped at the door.

A busy factory probably has its doors open all day, yet the movement of goods in and out of it is carefully controlled.  Foremen watch the crews and the doors to make sure the right shipments leave at the right times.  New shipments that arrive are confirmed before they are brought in.  Securing the movement of the goods is a constant, involved process.

With a cell, the conduction of goods in and out is much the same.  Water, oxygen and carbon dioxide can pass through the cell membrane freely, but many important molecules and ions need to be allowed in – or even brought in – by the membrane.  Otherwise, they can’t enter.  With exiting, it’s the same situation.

The cell “decides” what to let in at any given instant based on what it needs to maintain the composition of its cytoplasm.  The cytoplasm is the nonspecific, jelly-like “body” of the cell.  It’s a complex mix of water, ions, proteins, lipids, sugars, and other biological compounds.  Cytoplasm’s precise mix comprises the “living condition” of the cell.  If the mix goes wrong, the cell dies.  Cytoplasm can only be found in a living cell.

Therefore, the cell membrane maintains the “living” mix of the cytoplasm by controlling what enters and leaves the cell.  It’s a function that may not sound specific at first, but might be the most important of any the cell does.  Just how the cell membrane manages its role will be discussed in future posts.

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

Apostrophes: a minefield of everyday English

One issue in English tutoring is apostrophes.  We’ll sort out when to use them – and when not to.

In everyday English, you commonly see apostrophes misused.  There are only two cases in which you do use them:

1)  In contractions, such as can’t and don’t.  The apostrophe means that letters have been left out to shorten the word.  (For example, can’t means cannot.)

2)  To show possession, such as John’s car or Heidi’s boots.

You don’t use apostrophes any other time.

Note that, when the possessor is plural, you put the apostrophe after the s:

That is my parents’ car.

Hope this saves you some marks :)

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

The Tectonic Plates: Floating Islands of Rock

Tutoring high school science, you might be asked about plate tectonics at any time.  Here is the most condensed, pragmatic explanation you’ll likely find.

To believe in plate tectonics – which, by the way, is true beyond any doubt – you need to picture this:  Earth’s surface  consists of large islands of solid rock floating on a sea of magma.  There are about 9 large plates, plus some smaller ones.  Examples are the Indian Plate, the North American Plate, the South American Plate, the Eurasian Plate, the African Plate and the Pacific Plate.

The plates more or less cover the sea of magma beneath them.  At the same time, because the plates float on the magma, they move around like toy boats in a bath tub.  When they move apart, you have what’s called a divergent boundary.  When they crash together, you have a convergent boundary.  When they rub alongside each other, going in parallel but opposite directions (perhaps like people passing each other in a crowded hallway), you have a transform boundary.  Essentially, the term fault can be substituted for boundary.

At a divergent boundary, a crack eventually forms between the plates, then magma leaks up between them, so you get volcanoes.  The volcanoes can eventually form high ridges as they continue to erupt.  Divergent boundaries are more common under the sea than on land.

At a convergent boundary, there are two possibilities.  One is a head-on collision, in which case the plates buckle as they crash (like in a car crash).  The plates deform upward, resulting in mountains.  The Himalayas are such a mountain range, caused by the collision of the Indian Plate and the Eurasian Plate.

The second possible outcome at a convergent boundary is subduction, which is where one plate rides over top and the other slides below.  If one plate is a continental one and the other is a sea plate, the sea plate slides under and the continental one rides up, forming a mountain chain.  The Andes Mountains exemplify such a situation:  there, the Nazca Plate is sliding under the South American Plate.

Although each type of boundary hosts earthquakes, a transform boundary results, especially, in earthquakes.  As the plates try to shove past each other, they get stuck.  The pressure builds, then they break loose, resulting in sudden movement – and an earthquake.  The famous San Andreas Fault at San Francisco is a transform boundary: the Pacific Plate is running NNW, while the North American Plate is running SSE, along it.

Hope this gets you started :)

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

I pulled this article together from several high school textbooks:

Science Probe 10, Nelson Edition, Nelson Canada:  1996.

Earth Science, Spaulding and Namowitz, McDougal Littell:  2003.

The Changing Earth, McDougal Littell Science, McDougal Littell:  2005.

Longitude and Latitude

When you tutor social studies or geography, you’ll likely have to explain the concepts of longitude and latitude.  Now, we will.

The other day a kid came to me, embroiled in a conflict.  One adult had told him lines of longitude lie north to south, while another one had told him “north and south is latitude.”  Understandably, he was confused.  What’s more, kids have a way of believing the adult they last talked to, rather than the one in front of them presently.

Luckily, I wasn’t either of the two adults he’d already talked to.  Therefore, it was easy to explain to him that both those adults, in fact, had been right.  As so often happens, he thought they’d been telling him opposing views, when really they’d been telling him the same truth but in different ways.

As the first adult said, lines of longitude do lie north to south.  However, they don’t measure how far north or south you are.  How far north or south you are is measured by your latitude, as the second adult pointed out.  Of course, lines of latitude lie east to west.

“Think of a football field,” I told the kid.  “The yard lines lie sideways across the field, but they don’t measure how far sideways you are.  Rather, they measure how far forward you are.  They lie sideways, but they measure your forward position.  If you’re at 80 yards, it means you’ve crossed all the yard lines up to 80.  Running forward, you cross them because they lie across your path rather than parallel to it.”

It’s the same with longitude.  If you’re at 30° East, it means you’ve crossed the longitude lines from 1ºE through 29ºE, to land on the 30th one.  Going East, you cross those lines of longitude because they lie North to South.  If they ran East to West, then going East, you’d just stay on the same line forever.

In a similar way, lines of latitude lie East to West, but they measure your position north or south.  Note, for example, the Equator:  it’s at 0º latitude, yet obviously it runs East to West around the globe.

Although these ideas are obvious to anyone familiar with maps, they can be tough to grasp at first.

Well, the story has a happy ending.  After explaining longitude and latitude to the kid, I told him to find the position of Moscow for me.

“Around 37E, 56N”, he reported five minutes later.

Good enough:)

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

Algebra: solving equations with decimal coefficients

When you tutor math, decimal coefficients surface sometimes – especially in word problems.  We’ll sort out a simple scenario.

Recently, someone asked me a question very similar to

(1)   \begin{equation*}206.5(0.7x) + 112x = 4000.5\end{equation*}

Here’s what to do:
1)  multiply the 206.5 by 0.7x:

(2)   \begin{equation*}144.55x + 112x = 4000.5\end{equation*}

2)  combine the like terms on the left side:

(3)   \begin{equation*}256.55x = 4000.5\end{equation*}

3)  of course, divide both sides by 256.55:

(4)   \begin{equation*}\frac{256.55x}{256.55} = \frac{4000.5}{256.55}\end{equation*}

4)  we have the answer:

(5)   \begin{equation*}x=15.59345157\end{equation*}

Hope this helps:)

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

French: present tense of avoir and être

Whether tutoring French or learning it, these verb conjugations are critically important.

The French verb to be is  être.  Its present tense conjugation is as follows:

je suis (I am) nous sommes (we are)
tu es (you [singular] are) vous êtes (you [plural] are)
il est (he is) ils sont (they [masculine or mixed] are)
elle est (she is) elles sont (they [females only] are)

The French verb to have is avoir. Its present tense conjugation:

j’ai (I have) nous avons (we have)
tu as (you [singular] have) vous avez (you [plural] have)
il a (he has) ils ont (they [masculine or mixed] have)
elle a (she has) elles ont (they [females only] have)


Nous sommes dans le salon.    We are in the living room.

J’ai une gomme.    I have an eraser.

Not only are these conjugations important for everyday communication, but also they comprise the auxiliary for the passé composé – the French past tense.  We are headed in that direction – among others – for future posts.

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