8:00 -- Second period is an Algebra 1B class. This class is on the very same unit as yesterday's class, namely the laws of exponents. This time, there is a Pizzazz (actually "Punchlines") worksheet for this Algebra 1B class. (Yesterday, only Pre-Algebra had a Pizzazz worksheet.)
Most students figure out most of the problems, but the final problem is tricky because it actually contains variables in the exponent. I show the students how to do it and reassure them that this problem will probably not appear on any test.
8:50 -- Second period ends and tutorial begins. Several students show up this time.
9:25 -- Tutorial ends and third period begins. This is the other Algebra 1B class.
10:30 -- Third period ends and snack begins.
10:50 -- Like many special ed teachers (though not yesterday's!), today's regular teacher co-teaches another class this period. It's a Geometry class. These students are just starting Chapter 4 of the Glencoe text, which is on congruence and properties of triangles. In other words, they're slightly ahead of where we are currently in the U of Chicago text.
11:40 -- I return to my own classroom for fifth period. This is also a Geometry class -- except it's basically Geometry A for special ed students. The students take a quiz on Lessons 2-3 and 2-6 of the Glencoe text. These lessons are on conjectures, logic, and basic proofs. Again, Chapter 2 of the Glencoe text has much in common with the same numbered chapter in the U of Chicago text.
With the students about to take a quiz, I fear that a few students will mess around just as they did at the other school on Halloween. Fortunately, the aide makes sure that they are quiet. There is only one problem student in this class today.
Here's what happens -- one girl (who by the way was tardy) finishes her test quickly. Then she asks the aide for a restroom pass, and the aide lets her go. But then she's out of the classroom for about twenty minutes. Meanwhile, the aide only wants to let two students out of the classroom at a time, but then a third student asks to go to the restroom. Meanwhile, another teacher calls the room and summons this same girl in order to speak to her about something.
The aide walks out and searches for the missing girl. When she finds her, she escorts her to the other teacher to take care of business, then returns the student to our class.
Since this girl finishes her test early, this means that she must have aced the test, right? She already knows the material and is bored after the test, so that's why she asks for a restroom pass so she can take a breather after earning her "A," right?
Well, that's not exactly the case. I glance at her test and see that she's answered only one question correctly, namely the first one:
Make a conjecture about what the next number in this sequence would be: 1, 4, 7, 10, 13....
Meanwhile, she answers "true" to the following question:
Determine whether the following statement is true or false. If it is false, write a counterexample.
Any four-sided shape is a square.
And for the following question:
Rewrite the following statement as a conditional, converse, and
A polygon with three sides is a triangle.
I couldn't tell what she writes, but the word "if" doesn't appear in any of her answer. In other words, this student strolls in late, rushes to finish the test, gets only one question late, and then heads to the restroom for 20 minutes. In other words, attending every second of precious class time and trying to get a grade of "A" on today's quiz clearly aren't her priorities. Instead, her only priority is to get as much non-academic free time as possible, even if she hasn't mastered the material.
Notice that contrapositives were originally included on today's quiz, but were dropped. Lesson 2-4 of the U of Chicago text likewise teaches only converses, not contrapositives.
There are two questions that I don't like on today's quiz. The seventh and final question is:
Write a two-column proof.
Given:
Prove: x = -2.
This question introduces two-column proofs using Algebra I. This is common in many texts -- indeed the U of Chicago text hints at this in Lesson 3-3.
The problem is that we're to prove that x, which is supposed to represent a length, is negative. Of course, ST isn't x, but x + 2 -- but then x + 2 = 0, which still isn't a valid length. Notice that PQ is equal to 4(x + 2), or four times the length of ST, and yet
(This happens at other parts of the Geometry course as well. For example, there might be a question where the three angles of a triangle are given as algebraic expressions. We can solve for x and find the three angles, only to find out that an angle drawn as acute turns out to be 100 degrees.)
The other question is a bonus or Extra Credit Question. It asks for the inverse of "a corgi is a dog," after explaining what an inverse is. To the regular students learning about conditionals, converses, and contrapositives in the main text, it's natural to wonder what the fourth possibility (negating the hypothesis and conclusion without reversing them) is. But if contrapositives are dropped, then most likely negations are dropped, so students might be confused about what an inverse is. In the U of Chicago text, negations, inverses, and contrapositives are all delayed to Chapter 13.
12:35 -- Fifth period ends and it is time for lunch. Once again, sixth period is the teacher's conference period, and he is yet another football coach for seventh period. So my day essentially ends here.
Lesson 5-6 of the U of Chicago text is called "Alternate Interior Angles." The modern Third Edition of the text covers alternate interior angles slightly earlier, in Lesson 5-4 (along with Same Side Interior Angles, which aren't emphasized in the Second Edition).
This is what I wrote last year about today's lesson:
Today's lesson focuses on proving the Parallel Consequences -- that is, statements of the form, "if two lines are cut by a transversal, then ..."
This lesson will be set up almost exactly like Dr. Franklin Mason's Lesson 4.4. We wish to prove the converses of the Parallel Tests. We do so by using my favorite trick for proving converses -- we use the forward theorems along with a uniqueness statement. The uniqueness statement we need is the Uniqueness of Parallels Theorem -- in other words, Playfair.
It's possible to prove all of the Parallel Consequences by using the respective test plus Playfair -- so we'd prove the Corresponding Angles (CA) Consequence using CA Test plus Playfair, the Alternate Interior Angles (AIA) Consequence using AIA Test plus Playfair, and then Same-Side Interior Angles Consequence using that test plus Playfair.
But Dr. M only proves one of the consequences using Playfair -- he then uses vertical angles and linear pairs to derive the other consequences, as is traditionally done. We've seen that students should definitely be familiar with using one of the consequences to prove the others. Dr. M uses the Alternate Interior Angles Consequence to prove the others, but as we've discussed before, I'm changing this to the Corresponding Angles Consequence instead.
I looked back at last year's lesson and compared it to what I'm doing this year. In my first lesson after posting the Fifth Postulate last year, I posted some properties of two types of quadrilaterals, isosceles trapezoids and parallelograms. This year, we've already proved the isosceles trapezoid properties.
And now here's the activity I created -- a little something on spherical geometry (which is alluded to in Lesson 5-7):
Since I cut off the second part of the worksheet, I wish to replace it with something. I notice that in the U of Chicago and many other texts, students are given a taste of what would happen if we didn't have a parallel postulate. Yes, we'd have non-Euclidean geometry.
We've spent so much time on the blog discussing what's possible in neutral geometry and what requires a parallel postulate. The U of Chicago text introduces the students to spherical geometry -- as did I over the summer on the blog. But note that spherical geometry is not neutral -- neutral geometry includes only Euclidean and hyperbolic geometry. Technically, if we want to show students what impact the parallel postulate has on geometry, we should be showing them hyperbolic geometry, not spherical geometry. But since we live on a globe, spherical geometry is far easier for students at this age to understand.
Indeed, I often point out that the Fifth Postulate and its equivalents are, believe it or not, true in spherical geometry! This is because most of these statements begin with, "if two parallel lines..." or "if two lines are parallel." But in spherical geometry there are no parallel lines, so all Parallel Consequences are vacuously true! We might as well replace every occurrence of "parallel lines" with "unicorns," and all of the statements are still vacuously true:
Perpendicular to Parallels: If a line is perpendicular to one unicorn, it's perpendicular to the other.
Transitivity of Parallels: If a line is parallel to one unicorn, it's parallel to the other.
Parallel Consequences: If two unicorns are cut by a tranversal, corresponding angles are congruent.
Playfair: Using the version of Playfair given in Lesson 13-6 of the U of Chicago text doesn't work, but it does if we use Dr. Franklin Mason's version: "Through a point not on a given line, there's at most one line parallel to the given line." The phrase at most one allows for the possibility of zero.
Euclid's Fifth Postulate: If the two same-interior angles add up to less than two right angles, the lines will intersect on that side. This is true in spherical geometry because all lines (great circles) will intersect on both sides no matter what the angles add up to. Euclid's Fifth says nothing about what happens when the angles do add up to two right angles, only when they don't.
But still, I will mention spherical geometry as an example of a non-Euclidean geometry in which parallel lines don't work the way we expect them to.
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