In Algebra II, I pass out one last Pizzazz worksheet, Number 168. (Hmmm, perhaps I should have given them Number 169 instead on this Friday the 13th, since 13^2 = 169.) This worksheet returns to trigonometry, as students are given a trig table and asked to find simple values (such as sin of 35 degrees), as well as how to use them with triangles (if angle A = 40 degrees, then a/c = ...). The question is, "What Did The Leopard Say After Lunch?" and the answer is, "That really hit the spot!"
I believe it was Dr. Franklin Mason who believed that trig tables should be used instead of calculators, because students actually learn more about the functions by using tables -- for example, that they are not linear. (But unfortunately I can't find the exact quote online!)
During the third period conference, I cover a class called "Coordinated Math." This class appeared to be for students who are below grade-level -- the class consists of juniors and seniors, but the material looks a bit like Integrated Math I. The first question is to graph the image of square B (shown on a coordinate plane) after the translation (x, y) -> (x + 6, y - 3), and list the coordinates of the image.
Some students say that they want to work independently on the worksheet, rather than listen to me demonstrate the problems for them. But I show them how to do a few problems anyway. At the end, the same students who wish to work independently say that they do not understand how to answer even one question. The way that they make these comments implies that this is a group of students who believe that they are entitled to a math-free day, and will say anything just to let me give them a free day. And I have been looking forward to teaching a lesson on the Common Core transformations just to see whether I can explain them in a way that students will understand them (and also to see whether it's a good idea for transformations to be taught under Common Core in the first place). But if the students are determined to have a free day, I can't find out how much they really understand, or how effective my lessons are.
But if you recall, this is the reason I invented the Conjectures/"Who Am I?" game in the first place -- to motivate students to work who would otherwise treat themselves to a free period. My usual policy is that I only play "Who Am I?" with sophomores and below, as juniors and seniors should be mature enough to work on their own (and may find my game to be too immature). In fact, the first time I ever played "Who Am I?" with juniors was this week in Algebra II class -- only because I wanted the game to break up the monotonous block period.
But these juniors and seniors are below grade level students -- they are taking a freshman class, so I should have treated them like freshmen. I'll never know whether "Who Am I?" would have motivated more than half the students to work -- it's possible that those same students who say they want to work independently really would have done so (that is, they might have been several questions ahead of the game in order to earn more points)! Also, note that "Who Am I?" is a bit awkward with graphs, so I might have played the game only with the non-graph answers such as "find the coordinates" (though graphing it will help the students find them).
I notice on the worksheet that some of the questions direct the students to find the transformations using matrices. Of course, since I can't get many of the students to work on the non-matrix problems, I certainly have no idea how well they will do with matrices.
Chapter 18 of Morris Kline's Mathematics and the Physical World is called "The Mathematics of Oscillatory Motion." To oscillate means to go back and forth.
"Ut tensio, sic vis." -- Robert Hooke
Kline begins,
"The motions of projectiles, of planets, and of light have an obvious importance or attraction, and one would therefore hardly question why men should study these motions. On the other hand, the motion of an object suspended from a spring and bobbing up and down seems to offer no more attraction than to while away an idle hour."
The opening quote is Latin for "as the extension, so is the force." In algebra, we write this with the famous equation F = kd. But who is this Robert Hooke, the author of this quote? Well, let's ask Kline:
"The first of these [phenomena, the motion of a bob on a spring] attracted the attention of Robert Hooke (1635-1703), a contemporary of Newton, a professor of mathematics and mechanics at Gresham College in England, and a noted experimentalist."
And the key function of any oscillatory motion is the sine wave -- which goes right back to the sine functions that Algebra II students are learning about now. Kline describes in detail the equation:
y = D sin FA
where D is the amplitude and F is what Kline calls "the frequency in 360 degrees." Kline also introduces radian measure at this point, and so he writes:
y = D sin Ft
has a period of 2pi units of t. Of course, a full explanation of how to get from F = kd (often written with a negative sign, F = -kd) to the sine wave requires Calculus.
Question 18 of the PARCC Practice Test contains two proofs:
18. In the figure, p | | s. Transversals t and w intersect at point L.
(Here is some more info from the diagram: Line p intersects t at J and w at Z. Line q intersects t at H and w at K. Lines s, t, and w are concurrent. There are also three numbered angles. Angle 1 is at the northwest intersection of p and t, Angle 2 is at the northwest intersection of s and t, and Angle 3 is at the southeast intersection of s and t.)
Part A
Proof:
Statements Reasons
1. p | | s 1. Given
2. Angle 1 = 2 2. Corresponding angles along parallel lines are congruent.
3. Angle 2 = 3 3. ?
4. Angle 1 = 3 4. Congruence of angles is transitive.
What is the missing reason in Step 3?
A. Alternate interior angles along parallel lines are congruent.
B. Alternate exterior angles along parallel lines are congruent.
C. Corresponding angles along parallel lines are congruent.
D. Vertical angles are congruent.
Part B
Consider the proof of p | | q given that triangle LHK ~ LJZ. If triangle LHK ~ LJZ, then angle LHK = LJZ because corresponding angles in similar triangles are congruent.
Which statement completes the proof?
A. If angle LHK = LJZ, then p | | q because when base angles are congruent, the lines are parallel.
B. If angle LHK = LJZ, then p | | q because when corresponding angles are congruent, the lines are parallel.
C. If angle LHK = LKH, then p | | q because when alternate exterior angles are congruent, the lines are parallel.
D. If angle JLZ = HLK, then p | | q because when corresponding angles are congruent, the lines are parallel.
In Part A, we see from the diagram that angles 2 and 3 are clearly vertical angles, so the correct answer is (D). In Part B, the two angles LHK and LJZ are corresponding angles, so the answer is (B).
Notice that in Part A, the two angles to be proved congruent (1 and 3) are often called alternate exterior angles in some texts (though not the U of Chicago). This is relevant only because "alternate exterior angles" are mentioned in one of the wrong answer choices (B)! Notice that if we have an Alternate Exterior Angles Consequence Theorem, then we can shorten our proof greatly:
Proof:
Statements Reasons
1. p | | s 1. Given
2. Angle 1 = 3 2. Alternate exterior angles along parallel lines are congruent.
But since we are asked to supply the third step of a four-step proof, we must assume that AEA is not available to us, despite the mention of AEA in the answer choices.
In Part B, it should be easy to eliminate some answer choices -- (C) and (D) mention angles LKH and HLK, which have absolutely nothing to do with the previous step in the paragraph proof. If students know how proofs work, if the previous step mentions LHK and LJZ and there's only one step left to go, it's logical to assume that the final step has something to do with LHK and LJZ. The real problem is that many students don't know how proofs work -- especially not paragraph proofs like this one.
After we eliminate these two choices, we are left to choose between (A) and (B) -- "base angles" and "corresponding angles." Well, there is no Base Angles Parallel Test, but there is a Corresponding Angles Parallel Test, so the answer must be the latter. Notice that the angles mentioned in (C) actually are base angles of the triangle LHK. But we don't know that these base angles are congruent -- and if we could, the conclusion would be that LHK is isosceles, not that any lines are parallel!
PARCC Practice EOY Question 18
U of Chicago Correspondence: Lessons 3-2, Types of Angles
Key Theorem: Vertical Angles Theorem
If two angles are vertical angles, then they have equal measures.
Common Core Standard:
CCSS.MATH.CONTENT.HSG.CO.C.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Commentary: I included another Kuta worksheet from Integrated Math I on parallel lines and transversals, rather than the transformations assignment from Coordinated Math. I'd have preferred the latter since I want this to be an activity day, and a Kuta worksheet is a lousy excuse for an "activity." So I incorporated today's PARCC proof into Question 7 from the Kuta worksheet, since they have nearly identical diagrams. So the students have six warm-up problems before they reach the proofs -- which are no longer multiple choice. Notice that alternate exterior angles are actually mentioned on this worksheet. I listed "Vertical Angles Theorem" as the key theorem rather than the Corresponding Angles Test only because the latter is a postulate in the U of Chicago text rather than a theorem.
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