Chapter 5 Review, Continued (Day 59)

This is what Theoni Pappas writes on page 8 of her Magic of Mathematics:

"When the speed of air over the top of the wing is increased, the lift will also increase."

This is the last page of the subsection on flying. Pappas is in the middle of explaining how the angle between the wing and the approaching air -- called the angle of attack -- affects lift.

Here are some more excerpts from this page:

"If this angle increases to approximately 15 or more degrees, the lift can stop abruptly and the bird or plane begins to fall instead of rising. The angle of stall makes the air form vortexes on top of the wing. Not having been endowed with the flying equipment of birds, humans have utilized mathematical and physical principles to lift themselves and other things off the ground. Engineering designs and features have been continually adapted to improve an aircraft's performance."

The lone picture on this page is that of a glider flown by the Wright brothers.

I usually don't mention the footnotes on the page, but I couldn't resist mentioning this one:

"Laws governing the flow of air for airplanes apply to many other aspects in our lives, such as skyscrapers, suspension bridges, certain computer disk drives, water and gas pumps, and turbines."

In other words, the reason that buildings and bridges don't fall down on windy days is that they are built to withstand the wind -- and the same calculations used to make structures windproof are the same as those used to make planes fly. Again, we behold the power of mathematics!

This is our second review day, which means it's time to look around the web for an activity. The following worksheet comes from Elissa Miller -- yes, I know it's been some time since I last linked to Miller's website. This link is nearly two years old:

http://misscalculate.blogspot.com/2016/03/geometry-unit-6-quadrilaterals.html

As usual, these don't print well on my computer, so you may wish to go directly to the source. One thing about Miller's quadrilateral worksheets is that some of them mention parallelogram properties that don't appear in the U of Chicago text until Chapter 7. And so instead, I decided to post her trapezoid worksheet instead, even though the other quadrilaterals will appear on tomorrow's test.

Notice that in addition to "isosceles trapezoid," Miller uses the terms "scalene trapezoid" and "right angled trapezoid," neither of which appear in the U of Chicago text. Of course, we can derive their definitions from the analogous terms for triangles.

It also appears that Miller is using the exclusive definition of a trapezoid. For example, Trapezoid Problem #5 asks:

5. AB is not parallel to ____, ____, and ____.

The intended answer is BC, CD, and DA -- clearly AB intersects BC and DA, and since an exclusive trapezoid can't have more than one parallel pair, AB can't be parallel to CD either. But under our inclusive definition, we can't rule out the possibility that AB | | CD. Of course, the way these two segments are drawn, they don't appear anywhere close to parallel at all.