Monday, September 3, 2012

Labor Day: Reference Angle, Construction Calculators, Why a Negative Times a Negative is Positive

Reference Angle

Each angle can be represented by a reference angle. The reference angle is determined by finding the smallest angle between the terminal ("ending side") side of the angle and the x-axis.

The reference angle is always positive and has the range 0° ≤ θ ≤ 90°.

The reference angle of 135° is 45°.
The reference angle of 220° is 30°.
The reference angle of 329° is 31°.

This page by gives examples and a method of finding the reference angle.

Math warehouse also gives a good explanation of reference angle.

Another way to get a reference angle is to use a scientific calculator. For most, if not all scientific calculators and math programs, the arcsin function (usually labeled sin^-1) gives the angle between -90° and 90°. On a scientific calculator, we can use the following keystroke sequence:

Angle [SIN] [SIN^-1] [ABS]

Or the command abs(asin(sin(x)). A graph of the reference angle is shown below. The angle in the graph is measure of radians.

Construction Calculators

I usually accompany my dad to trips to Home Depot and Lowes whenever he gets ideas for home improvement. The latest one was to build a fence. My dad and I often talk about construction and home improvement plans.

Calculated Industries is a company that makes specialized calculators that cover construction, real estate, hot rods, electronics, and even cooking and quilting. While I dare say the math can be done with a scientific and finance calculator, these calculators are good because (1) the keys are specialized and (2) they focus on a specialized area.

I managed to buy all three of calculators the stores have, which two of them are pictured below. The Materials Estimator is just what it says, it's main focus is estimate the materials needed, such as of tiles needed, gallons of paint, or my favorite, the fence materials. The Construction Master 5 focuses on roofs, stairs, and circular structures. My favorite feature of the construction calculators is the ability to calculate dimensional math in yards, feet, and inches with fractions (precision can be set to 1/64).

Whether a home improvement project is actually going to happen I am not sure (cost is a factor), but at the least these calculators are nice additions to my collection.

Why a Negative Times a Negative is a Positive

We all heard the rule "A negative number times a negative number is a positive number". This line reminds me of the 1988 movie "Stand and Deliver". And yes, I had to look up the title because I remember the wrong name, but I remember Edward James Olmos as a teacher getting his classroom to chant the rule.

Confession: I am not a movie person, at all This may be strange considering I was born in Los Angeles blocks away from Hollywood. Sitting a dark room for two or more hours looking at a screen just doesn't do anything for me.

Anyway, I feel like for students to understand and fully grasp this concept, a nice short, concise explanation is needed.

Some resources tend to use an example such as:

-4 × -5
= -1 × 4 × -1 × 5
Using the Commutative Property for multiplication:
= -1 × -1 × 4 × 5
= -1 × -1 × 20
= 20.

Yet I feel this explanation is incomplete. There is no explanation of why -1 × -1 = 1.

I think a better way to show that a negative times a negative is positive is to consider the equation:

S = A × B + -A × B + -A × -B

Where S is the sum of the right hand expression, and A and B are positive numbers. (A > 0, B > 0).

We can find S in two ways. First, factor out a B in the first and second term of the right hand side:

S = A × B + -A × B + -A × -B
= (A + -A) × B + -A × -B
= 0 + -A × -B
= -A × -B

A second way is to factor -A on the second and third term on the right hand side:

S = A × B + -A × B + -A × -B
= A × B + -A × (B + -B)
= A × B + 0
= A × B

Note the following:
A > 0 and B > 0.
-A < 0 and -B < 0.
And A × B > 0, and S > 0.

Also, note S = A × B = -A × -B, and since S > 0, this implies -A × -B > 0.

There are many explanations that can be found on a web search either on a search engine or on YouTube. This one is my favorite because it is the most logical and it introduces logic and mathematical thinking to students.

That concludes this entry for today. Have a great Labor Day and I will talk to you soon!


This blog is property of Edward Shore. © 2012

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