# M1.4.8

**Writes an equation**to represent a problem situation.

### Proving the Pythagorean Theorem

A guided classroom exercise that contributes to the long-term development of creative thinking and analysis.

### Motivating Honest Budgets and Hard Work Simultaneously

Motivating honest budgets and profit expectations is difficult because some rnsubordinate managers will provide overly optimistic budgets to "look good" to rntheir bosses, while others will lowball budgets so they will "look good" when they rnexceed expectations. This paper explains how to motivate accurate forecasts of rnbudgets and profits.

### Finding All Solutions to a Puzzle

The non-zero digits can be organized into a 3-by-3 array such that each row, rneach column, and each diagonal adds up to the same total. How many solutions rnare there? Find and describe them.

### Thermo and Fluid Dynamics of a Homemade Lava Lamp

Lava lamps are just cool, that's all there is to it. Most everyone has spent at least an hour of their life just staring at a lava lamp. While the experiment outlined here isn't really the same as a commercial lava lamp it does demonstrate many of the important fluid and thermodynamic properties. Most importantly it provides students with an interesting and fun visual demonstration intended to motivate thought about the physics behind the experiment itself.

### Two Answers are Better than One

What powerful mathematical tool do you use everyday? If you have ever considered how much food you can eat and still leave room for dessert, or made a rough calculation of how long it will take to travel to school, you are using estimates to help guide and inform your decisions.

### Pilots, Airplanes, and the Tangent of Three (3) Degrees

When approaching an airport, pilots must learn to maneuver their aircraft visually, so that a stabilized approach to the runway can be flown at a constant approach angle. Precise approach planning insures a smooth transition to a landing within the Touchdown Zone (1) of the runway. Pilots must sometimes execute visual approaches that are varied in size, shape, and angle based upon a variety of factors such as: other aircraft, obstructions, noise abatement, or prevailing weather conditions. The standard approach angle however, is 3°.

### Isoperimetric Geometry

The isoperimetric theorem states that: "Among all shapes with an equal area, the circle will be characterized by the smallest perimeter" which is equivalent to "Among all shapes with equal perimeter, the circle will be characterized by the largest area." The theorem's name derives from three Greek words: 'isos' meaning 'same', 'peri' meaning 'around' and 'metron' meaning 'measure'. A perimeter (= 'peri' + 'metron') is the arc length along the boundary of a closed two-dimensional region (= a planar shape).

### Grandpa's Social Security

This simple example shows how algebra can be useful in the real world: Should Grandpa start receiving his Social Security Benefits at age 62 or should he wait until age 65?

### Can an Astronaut on Mars distinguish the Earth from its Moon?

Some day an astronaut will stand on Mars and look back at Earth. As Alfred, Lord Tennyson wrote Venus, Hesper, Were we native to that splendour or in Mars We would see the globe we groan in, fairest of their evening stars Could we dream of wars and carnage, craft and madness lust and spite Roaring London, raving Paris, in that peaceful point of light? Would we not, when gazing heavenward, at a star so silver-fair Yearn, and clasp the hands, and murmur: Would to God that we were there? But what exactly will one see from Mars?