It's a special ed class -- in fact, it's the same teacher whose class I covered the last time I subbed at this school -- October 17th. Thus you can refer back to my October 17th post in order to recall some information about this class. Again, I won't bother to do a "Day in the Life" for this special ed class (since I didn't do one for October 17th either). And yes, it seems as if the only time I sub at this school these days is for this class.
This district has an block schedule, with all classes on Fridays (which are also late days) so that the schedule aligns with days of the week. But the fact that there was no school in this district on Monday throws this week's schedule off a bit. To keep the number of odd and even days equal, when there's a holiday on Monday (or any day other than Friday), then the Friday schedule is changed to that of the missing day. Since Monday (had it not been MLK Day) would have been an odd day, today becomes an odd day. Thus it was a regular day, not a late day.
(I wrote about block schedules back in my Calendar Reform posts. All classes days are only necessary if there are an odd number of school days in the week -- if the length of the week is even, then a pure block schedule is possible. In the Gregorian calendar, the school week is five days, but only four days during certain holiday weeks. The Calendar Reforms I posted were based on there being four or six school days per week. But how holidays work in those calendars has yet to be completely decided.)
First period is observing (co-teaching) Algebra I -- which marks only the fourth time I've been in a math class in this district (with the first three being observing this same class)!
The students are taking for the Chapter 6 Test. As I mentioned back in October, this isn't exactly the same as the Glencoe text. In particular, Chapter 6 is on exponents and polynomials.
The resident teacher projects the following onto the front board:
- Review answers on Practice Test
- Study
- Collect IXL, Practice Test, and Notes
- Chapter 6 Test (26 multiple choice, 14 free response)
And during the test, she has me grade some of the work that she collects in Step 3. (This doesn't include the Notes, which she grades herself out of the students' Interactive Notebooks.)
Hmm, those three letters we see in Step 3 look familiar. That's right -- I used IXL software with my middle school students two years ago at the old charter school. I've written several times since then how I suspect my lack of success in IXL that year was because I didn't make the students write down their IXL work on paper. Considering that in today's class, IXL is something that is collected (that is, it's on paper), this only corroborates my suspicion.
Here's how today's IXL accountability paper works -- the teacher assigns five lessons for the students to complete on IXL. Then they must write down four questions from each section, for a grand total of twenty questions.
This is in stark contrast to how I taught IXL that year. I usually assigned one lesson at a time (which was more compatible with the Illinois State text, where only standard is covered per week). Since the questions on IXL increased with difficulty, this meant that students who answered 20 questions would enter the Challenge Zone, while today's students only answer the easiest questions in each of the five lessons. My method favored depth while this teacher's method favors breadth. (Assuming that all the questions are correct, my students would score 92 in one lesson while hers would score 36 in each of five lessons.)
I mentioned how another math teacher -- in the other district, with different software -- made his students write their computer answers on paper as well. And now I see another teacher doing the same, with IXL to boot. This suggests that I should have done the same. On the other hand, making and copying an explicit IXL accountability form worksheet would probably be overkill.
It's ironic that I'd be writing about IXL on January 25th, of all days. Exactly two years ago today, my old charter school dropped IXL in favor of SBAC Prep time. Still, today demonstrates how I could have used IXL in my middle school classes up until January 24th that year.
The other worksheet I graded was for Lesson 6.3 (of whatever text this happens to be). This lesson is called "Exponential Functions." It reminds me how the Common Core has pushed Exponential Functions (formerly an Algebra II topic) down into Algebra I.
I also grade papers for her second and sixth period classes, which are both Geometry. The assignment is a p-set from the text on parallelograms, rectangles, rhombi, and squares. I mentioned last week that in the other district, the second semester begins with parallelograms in the latter half of Chapter 6 of the Glencoe text. The same is apparently true in this district -- except that this is Lesson 7.4, not anything in Chapter 6. (Once again, today's school doesn't use the Glencoe text.)
Again, I don't know exactly what text this school is using. Notice that it could almost be the U of Chicago text, as parallelogram properties appear in Lesson 7-6. But some of the special properties of rhombi and rectangles appear much earlier, in Lessons 5-4 and 5-5.
Even though today isn't a late day Friday, there is something special today -- an assembly. Tomorrow night is the Winter Formal. The theme is 1970's disco, and the assembly is to announce the final candidates for this year's King of the Disco. First about six or seven members of the king's court dance down the main carpet in the gym, followed by the final three candidates -- the princes. Each young man is escorted by his mother or another older woman. Finally, last year's king arrives with his crown. He announces that voting will take place at lunch, and the winner will be announced between tonight's girls and boys basketball games.
In my other district, the high school assembly schedule essentially converts the "tutorial" period into an assembly period. Half the school would go into the gym during this period while the other would attend one of their classes. Afterward the two groups would switch. This school does the same, except that instead of tutorial (which this school doesn't have), it's "embedded support" (the last twenty minutes of each block) that's dropped to allow for assembly time. (I explained "embedded support" in an old 2016 post, where I was discussing various block schedules during the countdown to my first day at the old charter school with its 80-minute blocks.)
The other two classes are Academic Enrichment. In third period, one freshman girl from that same Algebra I class is unable to finish her test. (Normally, she would have stayed during embedded support time to finish the test, but that's been changed to assembly time.) The regular Algebra I test allows her to return during third period to finish the test. When she returns, I tell her that her homework assignment is a Pizzazz worksheet on integers to prepare her for the next chapter coming up. This is one of the hardest topics in Algebra I -- factoring.
In fifth period, I help one sophomore girl with her Algebra I assignment. She has a different math teacher who's slightly behind -- she's still doing the Lesson 6.3 worksheet on graphing exponential functions. So of course I help her out. She gets most of the questions right, except a few with fractions and decimals. I also tell her that factoring is coming up. I inform her that when I was a young Algebra I student, my teacher called factoring "the F-word" of math. But then again, my old class didn't teach exponential functions. (It could be that after being conditioned for over two decades that factoring is the most difficult Algebra I topic, that honor might now go to the previous chapter due to exponential functions.)
I'm glad that I was able to sub in my old district today. You might have noticed that this was a very light week of subbing in my new district. That's because in that district it's Day 90 -- in other words, it's the end of the first semester. Thus it's finals week at all high schools. We notice that substitute teachers are rare during finals week -- the regular teachers make extra effort to be in the classroom on the day of the big test. And so I take advantage by subbing in the old district where it's not finals week.
(Once again, notice that in Early Start districts, finals are before winter break, yet there are almost never a full 90 days of school before winter break. Indeed, I subbed in my old district on its respective Day 90, but it clearly wasn't finals week. And I mentioned earlier how Day 91 in the LAUSD was on Wednesday. In districts that don't give finals before winter break, more effort is made to make the two semesters mathematically equal, 90 days each.)
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
If the vertex angle of an isosceles triangle is 130 degrees, then each base angle is ____ degrees.
All we need to solve this problem are the Isosceles Triangle and Triangle-Sum Theorems:
x + x + 130 = 180
2x + 130 = 180
2x = 50
x = 25.
Therefore the desired base angles are each 25 degrees -- and of course, today's date is the 25th.
The only theorems we need to solve this problem are both in Chapter 5. In other words, this is the first problem on the 2019 calendar that we could actually assign our students the same day, since we're currently well past Chapter 5.
Lesson 9-8 of the U of Chicago text is called "The Four-Color Problem." This lesson doesn't appear anywhere in the modern Third Edition, because this is one of those "extra" lessons that we include mainly for fun.
In the past, I've mentioned several books and lectures which discuss the Four-Color Conjecture. One of these was David Kung's lectures. That day, I also posted a worksheet on reflections on the coordinate plane (not three-dimensional space), and so I'll repost them both today. I admit that it may be awkward to include planar reflections during the same unit that spatial reflection is taught (part of Lesson 9-5). But again, I want to prepare students for the PARCC and SBAC, and these Common Core tests are likely to ask questions about planar reflections over an axis. Unfortunately, this topic isn't fully covered in the U of Chicago text (at least not my old Second Edition, anyway).
This is what I wrote last year about today's lesson:
Then Kung moves on to a famous theorem -- the Four-Color Theorem. Just like Lesson 12-7 "Can There Be Giants" from yesterday, the Four-Color Theorem appears in the U of Chicago text -- in fact it's Lesson 9-8. We skipped over it only because we omitted Chapter 9 entirely (as the rest of the material, 3D figures, are covered more thoroughly along with volume in Chapter 10).
The Four-Color Theorem states that any map can be colored with at most four colors. The U of Chicago tells us that the theorem holds on either a plane or a sphere. Kung points out that on other surfaces, different numbers of colors are required. On a Mobius strip, six colors are needed. Both Kung and the U of Chicago text mention that for a special doughnut shape (or is it a coffee cup shape?) called a torus, seven colors are necessary.
CCSS.MATH.CONTENT.8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Notice that eighth grade is the first in which transformations appear.
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Notice that eighth grade is the first in which transformations appear.
Here are the two lessons that I'm posting instead. One is based on the reflections worksheet that the students worked on in class, and the other is based on the Four-Color Theorem -- another lesson inspired by Kung's lecture. Next week, Kung will continue with topology.
I wish to link to a member of MTBoS who actually teaches the Four-Color Theorem in class:
http://eatplaymath.blogspot.com/2015/10/the-four-color-theorem-and-pumpkin-time.html
Lisa Winer is the author of this post that is over two years old. She doesn't specify in what state she lives, nor does she make it easy for me to figure out what grade or class this is.
Anyway, in Winer's class, she uses the term "chromatic number" to describe the fewest number of colors required to fill in a map. The Four-Color Theorem, therefore, states that the chromatic number of any planar map is four. On a Mobius strip the maximum chromatic number is six, and of course on a torus the maximum is seven.
It's time to return to Euclid. Of course, he writes nothing about Four Colors or reflections across an axis, and so we proceed with the next proposition instead:
Proposition 11.
Propositions 11 and 12 are both constructions. Many of Euclid's propositions are constructions -- indeed, "The First Theorem in Euclid's Elements" (that is, Proposition I.1) featured in Lesson 4-4 is actually a construction.
Classical constructions are performed with a straightedge and compass, and David Joyce writes about the importance of actually proving constructions as theorems. But it's awkward to ask our students to perform a construction in three dimensions.
In this construction, we have a point A and a plane P, and we wish to construct the line perpendicular to P through A. How can our students do this? Is P the flat plane of the paper and A a point floating up in space?
It might be interesting to attempt Euclid's construction in the classroom. Here's how: We choose A to be a point on the ceiling and P is the plane of the floor. Thus our goal is to draw a point on the floor directly below A.
The key to this construction is to hang a rope from point A -- a rope that should be longer than the room is high. We can pull the rope at any angle and double-mark the points where the rope is touching the floor. I say "double-mark" because the point on the floor (where the rope touches) is marked (say with chalk), and then the point on the rope (where the floor touches) is marked (say with a piece of tape). The rope now can serve as a compass -- the point of the compass is at A, and the opening of the compass is set to the distance between A and the tape. The locus of all points on the floor that are the same distance from A as the point marked on the floor is a circle, and the locus of all points on a given line on the floor that are the same distance from A is a pair of points. So if we have a point (say B) drawn on a line on the floor, then we could find the unique point C on that line such that
All the lines on the floor can be drawn in chalk. There will be some plane constructions drawn on the floor as well, so we could use a large compass where the pencil has been replaced with chalk.
OK, so let's begin the construction. We start by drawing any line on the floor, and then we label any point on that line B. We now find C on this line exactly as given above -- we double-mark B on both the rope and floor, and then swing the rope to find C such that AB = AC.
Now we use the chalk compass to find the perpendicular bisector of
Then we double-mark D with a second piece of tape, and then find the point on the last line we drew (that is, the perp. bisector of
Finally, we find the perpendicular bisector of
Of course, this whole construction seems silly because of gravity. We can just hang a rope freely from A, label the point where the rope touches the ground F, and then we're done! The difference, of course, is that Euclid's three-dimensional space isn't physical space, and so there's no direction that's "favored" because of gravity or any physical force.
And so I'm not quite sure how David Joyce has in mind when he says he wants "the basics of solid geometry" to be taught better. Does he include Euclid's spatial constructions -- does he really want students to perform them? Or maybe he merely desires that students visualize the proofs in their minds while looking at the proof.
(Do you remember Euclid the Game, which is played on computers? Maybe in higher levels, players can make three-dimensional constructions that are difficult to perform in the real world!)
By the way, we can still modernize Euclid's proof:
Given: the segments and angles in the above construction.
Prove:
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2.
(Call it plane Q. In the classroom, Q is an invisible plane parallel to a wall.)
3.
4.
5.
6.
7.
8.
It might be tricky to reconcile this proof with the "rope" construction from above. In Euclid's construction,
But think about it -- given a point A and a line, how do we construct a line through A perpendicular to the given line? The answer is that, using the compass, we find points B and C on that line that are equidistant from A, and then find the perpendicular bisector (in that plane) of
But technically, all we really need is D, the midpoint of
In the end, let's just stick to the Four-Color Theorem and two-dimensional reflections. Here are the worksheets for today.
That's right -- today is an activity day, and this lesson naturally lends itself to an activity. There's no reason for me to post coordinate reflections today, but I guess I've always posted reflections and Four Colors together in past years, so I might as well keep both worksheets this year.
Oh, and before I end this post, I must remark that traditionalist Barry Garelick posted today. I've resisted the temptation to make a traditionalists' post this week. And my next regularly scheduled traditionalists' post isn't until Tuesday.
But let me predict what's going to happen -- over the weekend, several prominent traditionalists, perhaps SteveH and one other poster, will comment on the thread. And then I won't be able to resist responding to those comments here -- and I'll end up posting traditionalists on Monday, in addition to the regularly scheduled Tuesday post. That's how it always happens.
Since today's post isn't traditionalists, I won't even link to Garelick today. But I will say one thing today since it's related to today's subbing. Garelick writes about the virtues of traditional p-sets, especially those that gradually grow more difficult, which he calls "scaffolding." It's come to my attention that IXL satisfies that criterion -- the easy problems come first, and then they become increasingly hard as the student approaches the Challenge Zone. But I suspect that traditionalists like Garelick would automatically reject IXL only because it's computer-based. To them, only p-sets completed with only paper, pencil, and the printed text count as traditional.
But I suspect that students would be more enthusiastic about answering questions on the computer than out of a textbook. What I wish I did today was calculate what percentage of Algebra I students turned in the IXL assignment as compared to the percentage of Geometry students who turned in the traditional written p-set. It's too late to make that comparison now, though.
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