Calculator usage: negative numbers to rational exponents (with odd roots)

Tutoring high school math, scientific calculators are a permanent fascination. The tutor mentions a quirk many might share.

Due to

and the fact that a common exponent notation is

xa=x^a,

it follows that

(-8)^(2/3)=4.

Curiously, of all the calculators in front of me this moment, only two – the Casio fx-991ES PLUS C and the TI-83 Plus (a graphing calculator) – are willing to perform (-8)^(2/3) as presented. The others give “ERROR”.

Yet, invoking the rule shown above, we can rewrite the calculation as

3√(-8)2

Then, all the scientific calculators I have at hand will give the answer 4. However, the calculator aboard my mobile phone will only do so if the square is posed before the cube root.

HTH:)

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

Spreadsheets: Excel: the pi() function

Tutoring math, your curiosity naturally extends to spreadsheets. The tutor points out a neat feature of Excel.

If you type

=pi()

in a cell, the value of π will appear. I find that 14 decimal places are available.

Source:

www.math.com

www.engineerexcel.com

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

Math, engineering, trades: steepness measurement: definition of grade

Tutoring math, you encounter grade. The tutor discusses its definition and why it might be surprising.

Grade is defined as 100%*(vertical/horizontal). In the above diagram, it would be as follows:

grade=100%*(rise/run).

By itself, rise/run is called slope.

Therefore,

grade=100%*slope

What follows is a distinction that, to me, is important and interesting:

grade is not

rise/distance traveled

since, of course, you can’t drive along the horizontal course of a hill; rather, you can only drive on its surface.

At level, grade and (rise/distance traveled) are both zero. They remain virtually the same even at 20% grade, when (rise/distance traveled) is 19.6%. As the grade increases, however, they differ dramatically: at 100% grade, (rise/distance traveled) is 70.7%.

My interest in the difference between grade and (rise/distance traveled) is philosophical: why base a value on an indirect measurement (horizontal distance), when a direct measurement (distance traveled) is available?

In math, we use slope, of course; however, it’s usually in a context where actual measurements aren’t used. Rather, it’s just on paper.

Source:

engineeringtoolbox.com

connect.ubc.ca

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

Math: commutative operations

Tutoring math, you encounter definitions. The tutor defines commutative.

commutative adj
describes an operation for which the order of the inputs doesn’t change the output. Since 6×7=7×6, multiplication is commutative. So is addition.

Source:

Gallian, Joseph A. Contemporary Abstract Algebra. Toronto: D.C. Heath and Company, 1994.

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

Math: what does associative mean?

Tutoring math, you mention definitions. The tutor defines associative.

associative(adj):

independent of grouping: for instance, (a+b)+c = a+(b+c), meaning that addition is associative. Multiplication is associative as well.

Source:

Gallian, Joseph A. Contemporary Abstract Algebra. Toronto: D.C. Heath and Company, 1994.

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

Math: minimum sum of squares

Tutoring math, you encounter patterns. The tutor brings up one about sums of squares.

If two whole numbers sum to a constant even one, then their minimum sum of squares will come from each being half. The reason:

Let the numbers sum to 2n, and let the numbers then be n-k and n+k. Then the sum of their squares is

(n-k)2 + (n+k)2

Expanding each square, we get

n2-2nk+k2 + n2+2nk+k2=2n2+2k2

Therefore, the minimum sum of squares will be achieved when k=0, that is, when n-k=n+k=n.

Source:

Travers, Kenneth J et al. Using Advanced Algebra. Toronto: Doubleday Canada Limited, 1977.

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

Math: counting: how many ways can four whole numbers sum to 10?

Tutoring math, you might be asked counting problems. The tutor brings up one.

How many solutions can be found to x1 + x2 + x3 + x4 = 10, where x1, x2, x3, and x4 are all whole numbers?

Imagine three pipes |||, and ten asterisks **********. Then 5, for example, appears as *****. The pipes separate the values of x1, x2, x3, and x4. Therefore, the solution x1=3, x2 =0,x3=5, x4=2 shows as

***||*****|**

By the guidelines above, any solution to x1 + x2 + x3 + x4 = 10 can be shown as a sequence of 13 characters: three pipes and ten asterisks. The variability is which three positions the pipes occupy. 13C3 is how many ways the pipes can be distributed. Therefore, there are 13C3 solutions to x1 + x2 + x3 + x4 = 10, where x1, x2, x3, and x4 are all whole numbers.

Source:

Ross, Sheldon. A First Course in Probability. New York: Macmillan Publishing Company, 1988.

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

Psychology, probability: a parking game

Tutoring probability, you can imagine so many everyday examples. The tutor shares one.

This morning I went to pick up some groceries.

I like to load groceries through the rear door of the van. However, I also like to drive forward from a parking space, rather than backing up.

I arrived at the supermarket early, so there were many free parking spaces two deep: you could park in the first one or glide through to the second one, which faces an exit lane.

So here are the risks of the game:

  1. Park in the first row, forwards: you have definite access to your rear hatch but you will have to back out if someone parks in front of you.
  2. Park in the second row, forwards: you can definitely drive out forwards, but someone may park behind you, blocking your rear hatch.
  3. Park too close and the relevant possibility above (1 or 2) is much more likely; park further away and it takes longer to walk in and return (and I was, as always, in a hurry).

I opted for a fairly close spot and parked in the second row for guaranteed forward exit. However, I left two empty slots closer to the store. Would someone park behind me?

When I emerged, no one had parked behind me. However, someone had parked right beside me in the same manner as I, one space nearer the store. Obviously, they are willing to move a little closer to the edge.

Did someone park behind them? Who knows:) It’s interesting that they seem to calculate the situation very similarly to how I do:)

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

Math: working down the page vs across

Tutoring math, you encounter presentation methods. The tutor explains which is best.

At the high school level, math should be worked down the page, rather than across. Example:

2x – 7 = 11
2x = 18
x = 9

not

2x – 7 = 11 → 2x = 18 → x = 9

Reasons for working downward rather than across:

  1. For the student, tracking the progress is much easier.
  2. For the teacher, tracking the steps the student used is much easier.

I bring up that working downward is better, because this morning my wife told my older son, “Your father tells you to work down the page, and he’s absolutely right.”

My wife claiming I’m right is so rare, I think it’s newsworthy.

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

Statistics: how often can something “better” be expected to perform better?

Tutoring statistics, you might imagine everyday situations. The tutor brings up one.

Let’s imagine we have two mile runners. Runner 1, called R1, has mean time 4:45, with standard deviation 10s; R2 has mean time 5:00 with standard deviation 12s.

In any given race, give the probability R1 will beat R2.

Solution:

First, we convert the mile times to seconds: R1’s mean is 285s, while R2’s is 300.

The expected difference between R2 and R1’s time is 300-285=15.

We can’t add standard deviations, but rather variances: 10^2 + 12^2 = 244. The standard deviation of the difference is then 244^0.5 = 15.6.

The standardized statistic is z = (x-15)/15.6. We wonder p(x>0), which means p(z>-0.96). From the z-table, the answer is 0.8315.

So, R1 should beat R2 about 83% of the time.

Source:

Harnett, Donald L. and James L. Murphy. Statistical Analysis for Business and Economics. Don Mills: Addison-Wesley, 1993.

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