Chapter 5 Review (Day 58)

This is what Theoni Pappas writes on page 7 of her Magic of Mathematics:

"Since the bottom of this wing does not have this curve, the speed of the air passing under the wing is slower and thus its air pressure is higher."

This is the third page of the subsection "The Mathematics of Flying." Pappas is in the middle of explaining why wings are curved. As we can see, the air above the curved wing is moving faster with lower pressure, and the air below the wing is moving slower with higher pressure. And this leads to lift, which causes the plane to fly.

Here are some more excerpts from this page:

"The weight (the pull of gravity) is the vertical force that counteracts the lift of the plane. Drag and thrust are the horizontal forces which enter the flying picture. A bird creates thrust by flapping its wings, while a plane relies on its propellers or jets. The lift and gravity must be zero, while the thrust and drag must balance."

There are two pictures on this page, both of wings and the direction of the surrounding air. Here is the first caption, which explains in further detail the relationship between wing shape and lift:

"The wing's shape makes the distance over the top longer, which means air must travel over the top faster, making the pressure on the top of the wing lower than under the wing. The greater pressure below the wing pushes the wing up."

And here is the second caption:

"When the wing is at a steeper angle, the distance over the top is even longer, thereby increasing the lifting force."

The last sentence on this page is:

"Viewing birds swooping and diving reveals two other flying factors."

We won't read about these two other flying factors until tomorrow -- but as a hint, I'll say that it has something to do with the second picture -- wing angles.

Today is the review for the Chapter 5 Test. This is what I wrote two years ago about the worksheet for today's review:

There are many important concepts covered on this test -- isosceles triangles, quadrilaterals, and parallel lines. Here are the types of questions that appear on this test -- all of these coming from the SPUR section of Chapter 5 of the U of Chicago text.

• The first two questions direct the student to draw an example of the figure using ruler, compass, or protractor. I don't expect most classrooms to have a compass -- and if the students don't have a ruler and protractor, close enough is good enough. For students with a straightedge and compass, the necessary constructions are perpendicular lines (for the scalene right triangle) and congruent line segments (for the isosceles acute triangle -- a single arc is sufficient).
• Question 3 requires use of the Trapezoid Angle Theorem and algebra. Once again, this test will require the most algebra of any first semester chapter.
• Questions 4-5 require use of the Parallel Consequences and algebra. The first question is the Alternate Interior Angles Consequence while technically, the second one is actually Same-Side Exterior Angles. Savvy students should be able to figure it out by using, for example, the Linear Pair Theorem followed by Corresponding Angles. One way I remember is that any two angles formed by two parallel lines and a transversal are either equal or supplementary -- and one can eyeball it to see whether the angles are acute or obtuse to tell which ones are which.
• Questions 6-7 require use of the Isosceles Triangle Theorem -- neither one is straightforward, though, as either the Linear Pair or Vertical Angles Theorems are necessary. Once again, the students can look for acute and obtuse angles to find which ones are equal or supplementary.
• Questions 8-9 require use of the Triangle-Sum Theorem. The first one only needs arithmetic, while the second needs algebra.
• Question 10 requires use of the Quadrilateral-Sum Theorem. Once again, linear pairs, vertical angles, and algebra are needed.
• Question 11 requires use of the Quadrilateral Hierarchy Theorem. Students must arrange the various shapes from most general to most specific.
• In Questions 12-13, students must identify the quadrilaterals. The second one is a rhombus, not necessarily a square, since nowhere is it stated that the angles are right angles. Some might note that here I appear to contradict myself -- didn't I just say that in Question 5, students are supposed to assume that angle 1 is obtuse and angle 7 is acute in order to determine that they are supplementary, but here in Question 13 students should not assume that any of the angles are right angles? The difference is that in Question 5, identifying acute and obtuse angles is used as a mnemonic to remember actual theorems such as Linear Pair or Corresponding Angles Consequence, but in Question 13 nothing warrants knowing that there are any right angles.
• Question 14 is a true-or-false question about the Quadrilateral Hierarchy. If the statement is false, the students should draw a square that isn't a parallelogram. But they shouldn't -- because the statement happens to be true.
• In Question 15, students must determine whether the quadrilateral has any symmetry lines. As it turns out, the figure is a kite, so by the Kite Symmetry Theorem, it has a symmetry line.
• Question 16 is another true-or-false question about the Quadrilateral Hierarchy, but this time students are not directed to draw a counterexample. As it turns out, the statement is true -- that the bases have a common bisector is mentioned in the Isosceles Trapezoid Symmetry Theorem.
• In Question 17, students are to evaluate a conjecture. The U of Chicago text asks students to rank on an A-E scale whether they believe the conjecture to be true or false -- but I don't do this on a test, or otherwise every student would just choose C ("I'm not sure!"). So the students' only choices are true and false. As it turns out, the conjecture is true, and it can proved simply by knowing that half of 180 is 90. But students don't have to prove it -- proofs can wait until...
• ...Questions 18-19. The keys to the first question are the Alternate Interior Angles Test and the definition of trapezoid. The keys to the second question are two definitions -- those of circle and kite. Notice that the first proof underscores the preference for inclusive rather than exclusive definitions. Under the exclusive definition, EFHG could be a parallelogram -- and the way it is drawn, it almost looks like a parallelogram. We can't prove that it's a parallelogram -- but we can't prove that it's not a parallelogram either, which means that under the exclusive definition, we can't prove that it's a trapezoid either! The second question isn't a problem for the exclusive definition --OQPR is clearly nonconvex while all parallelograms are convex. But even if point O had been drawn outside of circle P, we could still prove that it's an inclusive kite, but it could be a parallelogram (hence a rhombus) and not an exclusive kite.
• Question 20 has the students draw part of the Quadrilateral Hierarchy. It goes without saying that inclusive definitions should be used, so a rectangle is an isosceles trapezoid.