Thursday, October 15, 2015

Review for Chapter 3 & 4 Quiz (Day 35)

Part VII of Benoit B. Mandelbrot's The Fractal Geometry of Nature is on randomness and consists of Chapters 21 and 22. This is a short section of the book, and Mandelbrot is mainly setting up for some real-world fractals that are seemingly random in the next section.

Yesterday, I mentioned that I wouldn't post last year's Tic-Tac-Toe construction today, as it made more sense to post it yesterday as part of a post-PSAT activity. So instead, we move on to the next thing that I posted last year at this point -- which happens to be review for a test.

This should be the last time this semester that I'm switching my quizzes and tests -- what I posted last year as a test is considered to be a quiz this year. But this raises another issue -- should I even bother to give any sort of assessment so soon after the PSAT -- not to mention last week's quiz, now relabeled as a test?

I sometimes wonder whether it is wise to give tests the same week as any schoolwide standardized test -- whether it's the PSAT, high school exit exam, or Common Core test. In particular with the Common Core test, on one hand, we want students to do well on the test, and so I have no problem with giving students light work the rest of the week. And besides, part of the SBAC here in California is the Performance Task, and so we have no choice but to spend class time on that Performance Task, not giving lengthy assignments or preparing for a hard classroom test.

On the other hand, we know that many older students don't take the Common Core tests seriously. So some of these may just answer all the questions randomly in a couple of minutes -- and then still have the audacity to claim that they were working so hard on the test that they should have little to no work in all their classes the rest of the week!

My philosophy is that I'd rather give such students the entire week off than risk convincing the students who might have worked hard on the test that they're wrong to do so or that they should focus on classwork instead. So I will attempt to lighten the classwork during the week of any major standardized test.

Of course, the PSAT is not the Common Core test. Still, many students know that the PSAT is not the real SAT, and so they may fill in random answers such as they would on a Common Core test. I more effectively send the message that they should work hard on the PSAT by avoiding multiple quizzes and tests for which they must study so close to the PSAT.

And of course, the grade level matters. If I were teaching Geometry to classes of mostly sophomores (or freshmen), I wouldn't break for the SBAC because students in those grade levels don't even have to take the SBAC. In a PARCC state, I would still have to break for the PARCC though.

(BTW, notice that some states are starting to use the SAT as the Common Core test that all juniors must take, rather than PARCC or SBAC. This most notably includes Michigan, which is switching from ACT to SAT for this purpose. In Michigan, sophomores now take the PSAT and freshmen the PSAT 8/9 in the spring as part of their state testing requirements. I'd like this plan if it reduced the number of standardized tests students have to take, but unfortunately states like Michigan make the students take end-of-course exams on top of the SAT or PSAT.)

So you can see why I might not necessarily want to post a quiz today. Recall that when I first posted this quiz and test last year, it was on the Early Start calendar, and so it was still late September when I posted them. It's unfortunate that when I pushed this back to the Middle Start calendar, the quiz and test landed now in mid-October, close to the PSAT.

What would I post today instead of the quiz? All of yesterday's discussion about Levels 7 and beyond almost makes me want to post more levels of Euclid: The Game instead. But there are problems with Level 7 that might make it unsuitable for a high school classroom.

Let's recall how I solved Level 7 earlier. Our task is to construct a line segment with the same length and same direction as AB but beginning at point C. Now I realized this meant finding the point D such that ABDC is a parallelogram -- and we just unlocked a parallel lines tool after Level 6. So I simply drew in AC and then used the parallel tool twice -- first to draw a line parallel to AB through point C, and then to draw a line parallel to AC through point B. In all, this required three steps -- one segment and two parallels -- which is a minimum solution.

But, as we found out, this construction is not valid in neutral geometry. We can't prove that CD is congruent to AB without a parallel postulate. In particular, the fact that the opposite sides of a parallelogram are congruent is a Parallelogram Consequence, but only the Parallelogram Tests can be proved without the fifth postulate. And therefore I didn't post Level 7 because I wanted to post only neutral constructions.

Then yesterday, we looked at the Second Theorem in Euclid's Elements, and we wondered whether we could use that construction to solve Level 7 instead. I will cut and paste the link for Proposition I.2., but first let me change the names of the points. The link calls the given point A and the given segment BC, while the game calls the given point C and the given segment AB. So I will change all the A's to C's and vice versa as I cut and paste the following information from yesterday's link:

It is required to place a straight line equal to the given straight line BA with one end at the point C.
Join the straight line CB from the point C to the point B, and construct the equilateral triangle DCB on it.
Produce the straight lines CE and BF in a straight line with DC and DB. Describe the circle AGH with center B and radius BA, and again, describe the circle GKL with center D and radius DG.
Since the point B is the center of the circle AGH, therefore BA equals BG. Again, since the point D is the center of the circle GKL, therefore DL equals DG.
And in these DC equals DB, therefore the remainder CL equals the remainder BG.
But BA was also proved equal to BG, therefore each of the straight lines CL and BA equals BG. And things which equal the same thing also equal one another, therefore CL also equals BA.
Therefore the straight line CL equal to the given straight line BA has been placed with one end at the given point C.
Q.E.F.

Now it's mentioned that four circles and two lines are required for the construction. This gives a total of six primitive steps -- and the minimum number of primitive steps for Level 7 is indeed six. It is noted that CB doesn't need to be drawn in order to put an equilateral triangle on it. But that equilateral triangle replaces only two of the circles, so that there would still be five steps required -- two more than our minimum. So it could be that it's impossible to reach the minimum without using the fifth postulate, or for that matter using only the first three postulates and Proposition I.1 (as Euclid does in his proof of I.2).

And so I end up with nothing to post other than the quiz review. Of course, teachers can still give this review without actually giving the test:




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