Math: how to graph a line

When you tutor math, you know that everyone from grade 9 up needs to know this.

Suppose you need to graph 2x - 7y = 21. There are two basic methods: the slope-y intercept method, and the intercepts method. Today, we will use the intercepts method: it’s quicker.

Example: Graph 2x - 7y = 21


We need to make a quick table of values, which we get by inserting 0 in one variable, then solving for the other. First, let’s “plug in” zero for x:

(1)   \begin{equation*}2(0) - 7y =21\end{equation*}

Since 2(0) is just zero, we arrive at

(2)   \begin{equation*}-7y=21\end{equation*}

Dividing both sides by -7 to find y, we get

(3)   \begin{equation*}y=-3\end{equation*}

So, when x=0, y=-3. We enter that result in our table:

x y
0 -3

Next, we ask about when y=0. Therefore, we plug 0 in for y as follows:

(4)   \begin{equation*}2x - 7(0) = 21\end{equation*}

Once again: 7(0) is just zero, so we continue:

(5)   \begin{equation*}2x=21\end{equation*}

Dividing by 2 on both sides, we arrive at

(6)   \begin{equation*}x=10.5\end{equation*}

So, when y=0, x=10.5. We write that result in our table:

x y
0 -3
10.5 0

Now it’s time to graph the result. From the table, we know that (0,-3) and (10.5,0) are points on our line. For a brush-up on how to graph points, see my post here. What we basically need to know is that to graph a point, we start from the origin (the center point), then go across the first number, up (or down) the second, and mark the point.

Once we’ve got the two points marked, we draw a line through them both. Like this:

Look for more coverage on graphing lines in future posts:)

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

Math: decimals to fractions

Tutoring middle school math, you get asked how to convert decimals to fractions.  The math tutor tells some obvious – and some less obvious – methods.

Converting a terminating decimal (one that ends) to a fraction is easy:

Step 1)  Notice how many digits are right of the decimal point.

Step 2)  Write the decimal digits on the top of the fraction. Then, on the bottom, write a 1 followed by the same number of zeros as there are digits to the right of the decimal point.

Step 3)  Reduce if necessary.

Example:  Convert 0.0015 to a fraction.

Step 1) There are four digits to the right of the decimal point.

Step 2) Initially, our fraction is \frac{15}{10000}. Notice that we need four zeros on the bottom, since there were four digits to the right of the decimal point.

Step 3) We reduce \frac{15}{10000} to \frac{3}{2000}. We get this answer from realizing that 5 divides into both the numerator and the denominator: 3 times into the numerator, and 2000 times into the denominator.

To convert a repeating decimal, there are a couple of shortcuts. Specifically, if a single digit is repeating, the equivalent fraction is that digit over 9. If a pattern of two digits repeats, the equivalent fraction is those two digits over 99.

Example: Convert 0.5555….. to a fraction.

Solution: The fraction is \frac{5}{9}.

Example: Convert 0.17171717……to a fraction.

Solution: The fraction is \frac{17}{99}.

For middle school purposes, the above techniques are probably enough. There is related content in my earlier blog article here. For those interested in reading further, please visit again:)

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

Biology: Three kinds of respiration

Tutoring Biology 12, you cover the cardiovascular system.  The biology tutor discusses the gas exchange aspect.

People commonly associate respiration with breathing.  However, from a biology point of view, the meanings are different.  Breathing is the physical process of bringing fresh air into the lungs and then pushing out “used” air.  Respiration means gas exchange.

There are three kinds of respiration:  external, internal, and cellular.

External respiration is the one everyone thinks of:  in the lungs, the blood drops off its carbon dioxide and picks up oxygen.

Internal respiration happens in the tissues.  Blood drops off its oxygen to the tissue fluid (whence it reaches the cells), while collecting the carbon dioxide that the cells are constantly producing.

Cellular respiration happens inside the cell, in the mitochondria.  It is the chemical process of burning glucose with oxygen to produce energy, carbon dioxide, and water. (The carbon dioxide produced by cellular respiration is, of course, what you breathe out when you’re running:))

Each of these aspects of respiration needs more discussion, but this is a good starting point.  Drop in again for more about them:)

Source:  Mader, Sylvia S.  Inquiry into Life, 11th edition.  New York:  McGraw-Hill, 2006

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

Math: Radians, Part II

Tutoring math, you deal with degree-to-radian conversions.  The math tutor continues on the topic of radians, picking up where he left off last July.

To know what a radian is, see my post here. In math 12, radians – “rads” – are introduced. In calculus, they are used almost exclusively.

As I earlier observed,

(1)   \begin{equation*}2\pi\ rads=360\ degrees\end{equation*}

It follows that

(2)   \begin{equation*}\pi rads=180\ degrees\end{equation*}

Dividing both sides by 6, we arrive at

(3)   \begin{equation*}\frac{\pi}{6}\ rads=\frac{180}{6}=30\ degrees\end{equation*}

Returning to the equation \pi\ rads= 180\ degrees, let’s divide both sides by 4 this time. We arrive at

(4)   \begin{equation*}\frac{\pi}{4}\ rads=\frac{180}{4}=45\ degrees\end{equation*}

For exact values, we have all we need for most conversions.

Example 1: Convert 210\ degrees to radians.

Solution: Realize that 210 = 7(30). Knowing that 30\ degrees=\frac{\pi}{6}\ rads, it follows that

(5)   \begin{equation*}210\ degrees= 7(30)\ degrees = 7(\frac{\pi}{6})\ rads=\frac{7\pi}{6}\ rads\end{equation*}

It’s really true: 210\ degrees=\frac{7\pi}{6}\ rads.

Example 2: Now, let’s go the other way: give \frac{3\pi}{4}\ rads in degrees.

Solution: We recall that \frac{\pi}{4}\ rads=45\ degrees. Then it follows that

(6)   \begin{equation*}\frac{3\pi}{4}\ rads=3(\frac{\pi}{4}\ rads)=3(45\ degrees)=135\ degrees\end{equation*}

Once again, it’s true: \frac{3\pi}{4}\ rads=135\ degrees.

In common usage, the unit “rads” is almost never written. Therefore, if an angle is given as 12 with no mention of degrees, you know it’s 12 rads.

The fraction-to-multiple method above is very quick once you remember the simple radian fractions and what they are in degrees. For most exact value conversions, this is all you need:)

In a future post I’ll be exploring the conversion between radians and degrees when it’s not an exact value.

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

Math: Constant Calculations on the TI-30XA

Tutoring math, you handle all kinds of calculators.  The math tutor continues with constant calculations:  this time, we hear Texas Instruments’ side of the story.

The TI-30XA is extremely similar to the calculator I used in high school twenty five years ago.  It’s such a good design, why change it?  Similar to the Casio fx-260Solar (mentioned in my last article), it’s reverse entry.  That is, to get the square root of a number, you enter the number first, then key for square root.  Same goes for trig functions:  to get sin30°, you enter 30 first, then press sin.

Recall from my last article that constant calculations are convenient when you’re doing the same operation with the same number repeatedly. For example, let’s say you’re converting masses in kilograms to weights in pounds, so you’re multiplying by 2.2 repeatedly. Let’s imagine the first mass you must convert is 30kg. You can enter 30 X2.2 2nd HYP =. Now, a K appears at the top right corner of the screen; it signifies that you have entered a constant. (Notice the yellow K above the HYP key.)

As the TI-30XA manual states: “a constant is a number with an operation.” Our constant is \mbox{multiply by 2.2}. Now, if you enter a number and then press =, the calculator will give the number multiplied by 2.2; entering 10 and pressing = gives 22.

Pressing ONC cancels the constant. Or else, you can just press any of the arithmetic keys (+,-,x, or ÷) to cancel the constant.  You’ll know when you have; the K will disappear from the top right of the screen.

For more calculator hints, please come back:)

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

Math: Using constants on the Casio fx-260Solar

Tutoring math, calculator use is perennial.  The math tutor introduces a nifty trick for the Casio fx-260Solar.

In physics, you often have a constant in a formula.  An obvious example is

(1)   \begin{equation*}F=mg\mbox{, where }g=9.8m/s^2\end{equation*}

High school physics students use the above formula hundreds of times.

There are a couple of ways to use the Casio fx-260Solar so that you don’t have to enter 9.8 each time. Here’s one:

Enter 9.8 then X X; doing so defines 9.8 as a multiplication constant. You’ll see “K” next to the “DEG” around top middle of the screen, meaning you’ve defined a constant. Now, pressing any number, then =, will give that number multiplied by 9.8.

As soon as you enter one of the arithmetic keys (let’s say, for instance, you enter 6 X 7), the constant is erased; the “K” next to “DEG” disappears.

In the Casio manual, this feature is referred to as “Constant Calculations.”

I’ll be saying more about calculator usage in future posts:)

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

Urine Regulation: Aldosterone

Tutoring Biology 12, you realize that with most organ systems, the hormonal control is the most difficult to retain.  The Biology tutor continues about urine regulation with this discussion of aldosterone.

Of course, urine is produced by the kidneys.  If you missed it, you can read how a kidney works in my post here.

In my previous post I talked about ADH and how the body uses it to regulate urine volume. There is another hormone – called aldosterone – that the body uses to control how much water is reclaimed from the filtrate. (Recall that the filtrate is the mix of water, ions, and small molecules first removed from the blood by the kidneys.)

Aldosterone is released by the adrenal cortex. However, the adrenal cortex needs to be informed to do so by the presence of renin in the blood. Renin is secreted by the cells of the juxtaglomerular apparatus, which are adjacent to the glomerulus and sense the blood pressure within. Specifically, when the cells of the juxtaglomerular apparatus sense that the blood pressure is too low, they respond by secreting renin into the bloodstream.

The renin circulates through the body to the adrenal cortex. Detecting the renin, the adrenal cortex responds by secreting aldosterone.

Aldosterone targets the cells of the distal convoluted tubule, telling them to let go of more K+ (K+ means potassium ions), but reclaim more Na+ (sodium ions) in compensation. The effect is that more water is reabsorbed from the filtrate, increasing blood volume and decreasing urine volume.

Unlike ADH, aldosterone does not result in blood dilution, since more ions are reclaimed alongside the extra water that is reabsorbed. Someone might ask, “If aldosterone increases the reclamation of sodium ions, how does that mean increased water reabsorption?” The answer is that sodium ions have a powerful pull on water – more powerful than potassium ions. So if you reabsorb sodium ions instead of potassium ions, more water will be drawn back into the blood as well.

Ultimately, the kidneys release renin – which leads to the release of aldosterone – in order to defend their own function.  If blood pressure is too low, the kidneys cannot filter the blood properly.  By increasing water reabsorption and therefore blood volume, aldosterone helps maintain the necessary blood pressure for proper filtration.

Source: Biology 12, Module 4: Human Biology 2. Open School BC, 2007.

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

Urine regulation: ADH

Tutoring biology 12, you cover kidney function.  The biology tutor introduces ADH, which is a hormone used to regulate urine volume.

For explanation of how a kidney works, see my post here.

Today, we focus on the fine tuning of urine volume. The hypothalamus monitors the concentration of the blood. It may decide, for instance, that the blood risks dehydration. How can the hypothalamus respond to help prevent dehydration?

The hypothalamus has the option of ordering the posterior pituitary to release ADH (anti diuretic hormone). ADH acts on the cells of the distal convoluted tubule and the collecting duct, causing them to be more permeable to water. The result is that more water will be reabsorbed back into the blood. Subsequently, blood volume will stay higher, while urine volume will decrease.

Let’s imagine the other situation: the person has just drunk lots of water to flush themselves out. In such a case, the hypothalamus will detect the surplus of water in the blood, so won’t order the secretion of ADH. The cells of the distal convoluted tubule and collecting duct will allow less water to be recollected, so more will be left in the urine. Urine volume will increase, while blood volume will decrease.

At night, the hypothalamus may order the secretion of ADH to keep urine acculumation low during sleep. The benefit: the person will not have to get up as often to urinate – or maybe not at all until morning.

Another hormone – aldosterone – can also be used to influence urine volume. It will be discussed in a future post:)

Source: Biology 12, Module 4, Open School BC, 2007.

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