Thursday, January 31, 2013

Using Geometric Formulas To Simplify Life.

Bring in a nicely wound cord or garden hose (see below)

Explain to your students that you want them to determine the length of the cord or garden hose with the only condition being that they cannot stretch it out; they must find another way to measure the length.

Notice that the wound cord or garden hose is simply a bunch of circles stacked on top of one another. What if we calculated the circumference of the top circle and multiplied it by how many circles are formed by the winding process? Certainly you will be dealing with a margin of error here but it will at least get you in the ball park.

Note: be careful not to count a circle more than one time.  

Tuesday, January 22, 2013

Area of Irregular Shapes Using Roman Tortoise Formation

Out of curiosity, I decided to estimate the how effective the Roman Tortoise Formation was against opposing archery fire. Having only a few minutes during my morning coffee, I decided to limit my estimation to simply the front of the Tortoise Formation. Such an exercise would be fun for a classroom as well. Here was my thought ‘quick-an-easy’ process, feel free to point-out something more scientific or any potential mistakes.

The Goal: To estimate the area prone to archery fire with the Tortoise Formation.

Given: Google tells me that the Roman shield Scutum had the following dimensions

Height of Shield: 42 inches

Width of Shield: 26 inches.

Average Height of Roman man was 5 feet, 6 inches

Converting to feet, here is a general idea of the shape from the reference point of an archer standing directly in front of the formation.

Noting the rectangular shape and multiplying the average height by the width of six Scutum shields gives us an approximate area of

5.5ft x 13ft = 71.5ft^2

A = 71.5ft^2

The obvious areas prone to archery attack are the legs below the shield and within the semi-ellipsis surrounding their heads. To estimate these areas we could subtract the height of the shield from the average height of the man (5.5ft – 3.5ft = 2ft) and multiply this by the width of the six Scutum shields.

2ft x 13ft = 26ft^2

L = 26ft^2

As for the area’s within the six semi-ellipsis surrounding their heads, we could use the width of the concave shields (2.17ft) as half the circumference of the ellipsis and approximate the height of the radius as .5 feet. This would give us an approximate area of 1.7 ft^2 per ellipse (multiplied by all six would yield 10.2ft^2).

E = 10.2ft^2

Thus, for an opposing archer standing in front of the Roman Tortoise Formation would have approximately

• Area Prone to Attack = Area below the shield (L) plus semicircle areas above their shield (E).
• Area Prone To Attack = L + E
• Area Prone To Attack = 26ft^2 + 10.2ft^2
• Area Prone To Attack = 36.2ft^2

Or, approximately 51% of their body is exposed.

**Note: This seems a little high so I might have made a wrong approximation in length somewhere. Also, the roman soldiers body does not encompass the entire exposed area, so one would need to account for that as well. Nevertheless, this was just for fun.

Thursday, January 17, 2013

Incorporating Puzzles (Homeschooling Our Two Boys)

Puzzles are excellent brain training and co-ordination improvement tools and are quite fun! In particular, they develop your abilities to reason, analyze, sequence, deduce, logical thought processes and problem solving skills. These types of puzzles also improve hand-eye co-ordination and develop a good working sense of spatial arrangements. See Full Article