In this district, today is Day 115, thus making it late in the second trimester. It's significant because in this district -- believe it or not -- middle school students must take trimester finals. For a teacher to have a sub during finals week is rare, but today's teacher is about to become a grandmother again, and so she's traveling up north to be with her pregnant daughter. (Babies don't know about finals week!)
There are only a few other middle schools I'm aware of with finals weeks. The school I attended as a young student -- which spanned grades 7-12 -- resisted having finals weeks for any grade levels for a long time. Within the past decade, the school implemented finals week for all grades -- and then promptly switched to starting school in mid-August to accommodate the semester finals! Meanwhile, it's arguable that the "Benchmark Tests" that the old charter school implemented three years ago were intended to be graded like finals (particularly the November Benchmarks).
Even though it's Day 115 (with the mathematical two-thirds point not until Day 120), having finals this week fits the period rotation better. At this school, all periods rotate. The first day of the second trimester (in December) started with first period, and then on subsequent days the rotation starts with second period, third period, and so on throughout the trimester until today started with sixth. Then tomorrow and Friday are the two finals days (three tests each day, unlike high schools). And then the new trimester starts on Monday with first period yet again. (On the other hand, Thanksgiving divided the first tri and second tri, and so the last day of first tri wasn't necessarily sixth rotation.)
As for the finals days themselves, tomorrow starts with second period, and then the periods go in order from there, ending with first period as the last test on Friday. (Why do we start with second? I think it's because it's second trimester.) Oh, and by the way, recall that last week, I subbed in a science class that had an upcoming district CER test. As it turns out, that science test is scheduled for the finals days (in lieu of a science final).
Now that we know the schedule, let's launch "A Day in the Life."
8:15 -- At this school, homeroom is the same as first period -- which is conference period. But I'm asked to cover another room for homeroom only.
8:30 -- As I wrote earlier, today starts with sixth period. This is one of the eighth grade math classes, and all math classes have an aide.
It goes without saying what the students are doing today -- preparing for the final. Here are some of the questions in the review packet:
14. 43t - 31 = 53t - 91
25. Are the figures below similar? Why or why not?
32. What is the slope of a line passing through points (3, -4) and (-2, -7)?
33. Create a table of values for y = x + 6.
52. The line graph show the centimeters of rain that has fallen each hour during a storm.
a) If the rate of rain continues, about how much rain will have fallen at 16 hours?
b) Is this linear?
c) Is this proportional?
I decide that since this is review for a major test, we can play Conjectures/"Who Am I?" today. So I divide the class into groups and start with the usual "Guess my age" and "Guess my weight." But too many students are talkative throughout the game and struggle with the questions. I fear that my age and weight questions (designed to motivate the groups into wanting to earn points) only waste valuable time that can be spent on the review.
When the review is complete, there is supposed to be another worksheet on solving equations (similar to #14 above). The aide passes this worksheet out with just a few minutes to go, and so I quickly sing "Solving Equations" while doing some of the problems.
9:20 -- Sixth period leaves. As I wrote earlier, it's first period conference, which leads into snack.
10:30 -- Second period arrives. This is the lone seventh grade World History class. It's also the only class without an aide.
This class doesn't have a final this week. (I don't know whether it's true for history classes in general or what -- all I know is this class.) Instead, they are currently learning about early Islam. There are pictures of historic mosques and other buildings posted around the room, and the students answer questions on worksheets about these pictures.
Since these worksheets are also part of packets, I'm justified in singing "The Packet Rap" as my song for this period, rather than "Solving Equations."
Despite a slow start to this class -- time is wasted because I end up having to check everyone's Chromebook individually for yesterday's vocab homework -- this is the second best class of the day.
11:25 -- Second period leaves. Third period is a co-teaching eighth grade math class.
This class is also finishing up the review packet. But this time, the regular teacher has the students work independently and just goes over the answers at the end. My students are mostly special ed, and so only a few are able to work independently.
Also, there are a few extra questions on these kids' review packets that have been omitted from the special ed review packets. Presumably, these students will have fewer questions on the final as well.
I notice that just like the history class, there are laminated sheets of paper posted in the room. But these pictures are actually "Whodunnit?" clues. These clues aren't for this Common Core Math 8 class though, but for the eighth grade Algebra I students. They are on the laws of exponents and simplifying monomials. If the "Whodunnit?" clues had been for the class I'm present for, I certainly would have sung the "Whodunnit?" song I wrote at the start of this trimester. Instead, I don't sing at all, especially since going over the answers lasts until the end of the period.
Near the end of the period, the resident teacher informs me that after the final, the Math 8 students will begin studying exponents -- in other words, it's what the Algebra I students just finished! Once again, this underlines how much overlap there is between Math 8 and Algebra I. (Well, at least she can use the same "Whodunnit?" clues if she wants.)
12:15 -- Third period leaves for lunch.
1:00 -- Fourth period arrives. This is the second of three Math 8 classes (not counting co-teaching).
For this class, I drop "Guess my age!" and "Guess my weight!" and go straight into review. But it doesn't help enough -- there are still too many problems with two-thirds of the class getting wrong. I end up simply doing most of #52 (seen above) on the board and have all students copy to earn a point (during the "this question is for everybody" stage of the game), which at least sets them up to answer the next question #53 (as a "race" between groups), which looks just like the previous question.
This class also starts off on the wrong foot as too many students ask for passes right after lunch, during the silent reading time. But since this class isn't quite as noisy, I name it the best class of the day despite so many wrong answers (that is, behavior trumps academics).
2:05 -- Fourth period leaves and fifth period arrives. This is the last of three Math 8 classes.
This class is a little stronger academically, but also a bit louder. Thus there are fewer students in the class whom I fear will fail the final, but more students I fear will talk during the final.
2:55 -- Fifth period leaves, thus ending my day.
Today is Fourday:
Resolution #4: We need to inflate the wheels of our bike.
This definitely comes into play today more than almost any other day, as students need to remember what they learned during the trimester in order to succeed on the final.
And so I tell the students about the "bicycle" and what it means to inflate it. Since slope appears on the final, Sarah Carter's "Slope Dude" is relevant here. And in any class at this level, at some point students need to remember integer rules for addition, subtraction ("keep, change, change"), and multiplication as well.
By the way, I look ahead to Monday. The aide shows me the packets that the students will work on next week after the final. They are on slope (even though they just learned slope for the final). And I notice that another Carter-ism appears in the packet -- HOY-VUX. (Yes, Sarah Carter -- your mnemonics are starting to catch on.)
And indeed, when I have these multi-day subbing assignments in math classes, I try to take advantage and prepare an original song (just as I did with the "Whodunnit?" song).
Meanwhile, all of this again reminds me of "Benchmark Tests" at the old charter school, and how similar they were to finals. (There were Benchmarks at the start of the year, but here I'm specifically referring to the tests at the end of the first trimester.) They were a huge headache that year -- mainly because they were suddenly sprung upon us with little warning. There was a school calendar at the start of the year that showed Benchmarks near the end of the first semester, not trimester. Also, the material to be covered on the Benchmarks was uncertain. Oh, and there was also an unexpected online component (via Study Island) to these Benchmarks.
Sometimes I wonder whether the tests would have gone more smoothly if only we'd known more about them in advance -- if, just like finals in my new district, there was a calendar that showed certain minimum days reserved for these tests. Then it would have been obvious that there were upcoming assessments that we needed to prepare for.
One problem was that my old charter school was K-8. Middle and high schools often have weeks of minimum days at the end of the term for finals, but elementary schools tend to have weeks of minimum days at the start of the trimester -- just after the old tri has completed, so that report cards can be given at Parent Conferences. Our K-8 school followed the elementary model, so that there were Parent Conferences at the start of second tri, not finals at the end of first tri.
One interesting solution would be to have the trimesters for the elementary grades end a week before those at middle school. Then the elementary Parent Conferences and middle school finals could have been the during the same week of school-wide minimum days. Some of those days might have been reserved for written Benchmarks, while others could be for Study Island.
Again, most K-8 schools don't do this, but if the school is going to insist that the Benchmarks be included in the middle school students' grades, then they might as well be honest, call them "finals," and then set up a finals week for them.
This plan would have had the advantage of eliminating Parent Conference weeks for middle school teachers (who wouldn't have such weeks at pure middle schools). It makes it clear that we teachers need to prepare our students for the finals. The first such week would have been November 7th-10th (a four-day week due to Veteran's Day), and the second such week would have been March 6th-10th (thus avoiding a minimum day on Pi Day and having school open on 3/14 at 1:59.) I suppose that the last week of school would have been third tri finals (but this was also eighth grade graduation week and "Week of Service" for all lower grades).
Of course, this doesn't work if it's never revealed to us what lessons are assessed on each final. And indeed, if during the year that I failed to teach much science, a science final was suddenly announced barely a week in advance, the students would have complained loudly -- and rightly so.
Speaking of which, this week's final in my new district covers Chapters 3 and 4 of the Big Ideas eighth grade math text. Chapter 3 is on angles and triangles (including similar triangles) while Chapter 4 is on slope. This follows the Common Core suggestion of using similarity to teach slope (impossible using the Illinois State text since all EE standards appear before any G standards). In fact, transformations, including dilations, appear in Chapter 2 of the text. (I believe that the final for gen ed students extends at least to Chapter 5 on linear systems, if not Chapter 6 on functions.)
This is significant for the blog, since we're currently in the similarity chapter of the U of Chicago Geometry text. Questions like #25 on similarity and #52 on proportionality fit during the current chapter (if not the current lesson). And in today's post, I cut-and-paste some discussion of how Common Core teaches similarity.
Lesson 12-4 of the U of Chicago text is on proportions. I wrote a lot about comparing the U of Chicago approach to those of other theorists, and I retain some of that discussion in today's post.
Let's start with Hung-Hsi Wu. His "Fundamental Theorem of Similarity" actually consists of some of the properties of dilations that appeared in yesterday's Lesson 12-3. If the image of
Wu proves this in cases based on what sort of number the scale factor k is. For natural number k, Wu proves it using induction on k. This initial case, k = 2, is based on a special version of the Midpoint Connector Theorem of Lesson 11-5. The inductive case from k to k + 1 involves repeating the Midpoint Connector Theorem argument over and over.
For rational number k, if k = 1/q (q natural number), then Wu notes that a dilation of scale factor 1/q is the inverse of a dilation of scale factor q. And if k = p/q (p, q natural numbers), then Wu notes that a dilation of scale factor p/q is the composite of two dilations, with scale factors p and 1/q.
All that's left is to extend the argument to irrational k. Wu hand-waves over this by using what he calls the "Fundamental Assumption of School Mathematics" -- many theorems of pre-college math that apply to all rational numbers also apply to all real numbers. (This assumption also appears in Algebra II when defining what it means to raise a number to an irrational power.)
Even though Wu gives this proof, it's not the sort of proof we expect high school students to figure out easily. In the years since I posted it, I've regretted it. But every year since then, when I return to Lesson 12-6, I keep the Wu proof and change other parts of the worksheet!
As I wrote in the comment above, the EngageNY curriculum is based on the Wu proofs. But there's no way in the world EngageNY would use Wu's "Fundamental Theorem of Similarity" proof.
Instead, they replace it with a simpler proof based on the area of a triangle. I've mentioned the idea before how in proofs, area and similarity are often interchangeable -- this is why the Pythagorean Theorem has both area and similarity proofs. EngageNY's area proof removes the need for Wu's induction on k and subsequent extension to rational and real values of k.
But there's one problem here -- similarity is a "Module 2" topic, but area is a "Module 3" topic. This is the naive order suggested by the Common Core Standards -- since they mention similarity before area, all similarity lessons must appear before any area lessons. Once again, EngageNY justifies this by having students recall the triangle area formula from eighth grade (or earlier).
Notice that in the U of Chicago text, area (Chapter 8) appears before similarity (Chapter 12). Thus the U of Chicago could validly follow the EngageNY sequence of proofs -- except that it doesn't.
Many traditionalists attack the Wu/EngageNY plan from not following Euclid's geometry. But ironically, Wu/EngageNY actually follow Euclid more closely than traditional texts do! His Book VI, which teaches similarity, begins with Proposition 1 (ratios between areas of triangles). This is followed by Proposition 2 (Side-Splitter Theorem), from which both Wu/EngageNY derive their first similarity theorem (Proposition 4, AA~).
Of course, there's also the idea of deriving the similarity theorems "classically" -- that is, by assuming AA~ as a postulate -- and then ultimately deriving the properties of dilations. This is the method used by the old PARCC test, which should be irrelevant since California isn't a PARCC state, and backlash against the Core has led to most PARCC states dropping that test. But I still end up posting old PARCC questions from time to time -- including the Lesson 12-3 worksheet from yesterday.
Here is today's worksheet.
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