Chapter 6 Test (Day 70)

Today is the third day of subbing in the high school Spanish class. As I mentioned yesterday, I won't do "A Day in the Life" today.

As of now, this is the last day of subbing in this class. But the lesson plans left by the regular teacher include Monday as well, as if he's considering taking Monday off as well. It could be interesting for me to sub on Monday too, since today we continue but don't complete McFarland USA in Spanish I.

In Spanish II, the students continue working on the review packets for next week's test. They're due on Monday, so the test won't be until at least Tuesday -- once again hinting at the idea that the regular teacher won't be back until then.

But for us in Geometry, today is the Chapter 6 Test. Believe it or not, this is my only test this year that's scheduled for a Friday. It just works out that way that the only day count that's a multiple of ten that falls on a Friday is today, Day 70. There will also be a test scheduled for the Thursday right before a four-day (Friday-to-Monday) weekend. On the other hand, there's also going to be two upcoming tests scheduled for Mondays.

This is what I wrote last year about today's test:

Let's look at these final four questions in more detail. Questions 17 and 18 are graphing questions, except that one is transforming triangles, not snowmen. One of them is a glide reflection, while the other is a translation. I just hope that students won't be thrown off by seeing rules for each of these transformations, such as T(x, y) = (x + 5, -y).

Question 19 is about the cardinality of a set, N(S), which is mentioned briefly in Lesson 6-1 of the U of Chicago text, but I only discussed it briefly this year. Here's what I wrote about N(S) last year:

I decided that the only real reason that the U of Chicago introduces the N(S) notation for cardinality (number of elements in a set, previous question) is to prepare the students for function notation, so I might as well use it here. There's only one other place where I see n(A) used for number of elements in set A -- the Singapore Secondary Two standards!

I also wrote: (A Thanksgiving reference! These are the seven dates in November which could be turkey day!) Again, I originally wrote this test at Thanksgiving. Okay, this year we can pretend that the seven elements of S correspond to Christmas Day and the three days before and after Christmas.

The final question shows one more transformation -- which happens to be a dilation. Neither last year nor this year did I formally cover dilations. This is supposed to be a think-outside-the-box question where students should try to reason out what's going on. But think about it for a moment -- the graph makes it appear that (3, 3) is the image of (1, 1). So all students have to do is plug in x = y = 1 into each of the four choices and see that only choice (d) gives (3, 3) as the answer. (Of course, this year, the students have seen dilations because of Tom Turkey and Thanksgiving again.)

Students might consider the last four questions to be unfair. But even if they get all four wrong, it's still possible to get 80% -- the lowest possible B. So strong students who completed the review worksheet yesterday should still earn at least a C on this test.

Here are the answers to today's test -- the same answers I posted last year to the invisible test:

1. a translation 2 inches to the left

2. a translation 2 inches to the right

3. a rotation with center O and magnitude 180 degrees

4. a translation 8 centimeters to the right

5. true

6. angles D and G

7. triangle DEF, triangle GHI

8. Reflexive Property of Congruence

9. definition of congruence

10. Isometries preserve distance.

11. translation

12. translation

13. glide reflection

14. glide reflection

15.-16. The trick is to reflect the hole H twice, over the walls in reverse order, and then aim the golf ball G towards the image point H". In #15, notice that y and w are parallel, so reflecting in both of them is equivalent to a translation twice the length of the course. In #16, notice that x and y are perpendicular, so reflecting in both of them is equivalent to a 180-degree rotation.

17. glide reflection (changing the sign of y is the reflection part, adding to x is the translation part)

18. translation

19. 7

20. d (for dilation, of course!)

Now today's a test day, and it's been a while since I posted a topic about traditionalists, so let's make this our traditionalist post.

None of our main traditionalists have posted recently. But several days of watching McFarland have me thinking about something related to the debate.

Today, we watched the middle third of the film -- starting right after the weekend invitational (McFarland finishes in fourth place) and ending right after the state qualifier (again McFarland finishes in fourth place), a good 40+ minutes of movie. (Of course, there's a big difference between these two races -- there are only four teams at the invitational, while the top four at the qualifier advance to the State Meet.)

The school-to-prison pipeline makes an appearance in this section of the movie. In fact, McFarland high school is right next door to a prison -- during the home dual meet on the McFarland course, the athletes run next to the prison. One teacher remarks that most students at the school are most likely headed for either the fields or the prison.

Many traditionalists disagree with the concept of the school-to-prison pipeline. For example, here's a link to Momof4 (a frequent commenter at the Joanne Jacobs who often leans traditionalist). She suggests that it's not a "pipeline" but student attendance that's the problem:

https://www.joannejacobs.com/2018/07/less-discipline-more-disorder/

Momof4:
In the linked Heriot article, I found the answer to a question I had posed on another website; whether students suspended/deserving suspension also are often truant – and they are. I suspected that was the case, from seeing the DC attendance data of last year. That undermines the whole “school to jail pipeline” meme; in which I have never believed, because common sense suggests that the same kids who do not obey school rules also fail to obey the law outside of school. That does happen regardless of the reason for their absence; just as it happens outside of school hours. Juvenile crime stats make this pretty clear.

Also during this section of the movie, there's a scene where Coach White is having all of his runners study for the SAT. Recall that many of these runners are "pickers" -- that is, they spend much of their spare time picking crops in the fields. Coach White informs them that they can go to college and make more money to support their families. They might even considering majoring in agriculture, since their knowledge of the fields might put them at an advantage.

But the father of one athlete (#1 runner Thomas Valles) disagrees with this idea. Now we need to backtrack a little to yesterday's section of the movie for you to understand the relationship between Thomas and his dad.

When Coach White first arrives at McFarland High School, he finds out that Thomas has been beating up other students. They've been making fun of his sister because she is pregnant. White makes a deal with Thomas -- he'll avoid a suspension for fighting if he joins the cross country team.

Senor Valles is an itinerant picker. He often travels to other states to find work. But he returns to California when he hears that his daughter is pregnant. (By the way, this is the first time I've made the connection between the first McFarland scene and this one -- both Thomas and his father act they way they do because they're both reacting to the young expectant mother in their family. That's what I notice when I watch the same movie four times in one week.)

Senor Valles is so upset that he slams his hand into the wall. Thomas is afraid that his father might injure his hand and be unable to work in the field, so he tries to intervene -- and ultimately he gets hit in the face himself.

So after Coach White tells Thomas and the other runners about the SAT, the young athlete brags to his father that he might go to college someday. His father warns him in Spanish that he'll ruin his eyes if he stares at books for too long -- and that there's no need for a college degree in the fields, where the youngster is expected to work for the rest of his life.

Let's examine this scene from a traditionalist perspective. We know that the traditionalists like to push classes such as eighth grade Algebra I and senior-year AP Calculus. Not every student needs to take higher math class because not every student is going to college at all, much less major in STEM. Yet the traditionalists like the idea of "keeping doors open" by taking higher-level courses. They'd rather a student take too many rigorous courses than too few. I wonder whether they can imagine a parent who would actively discourage the child from taking the SAT or higher-level classes.

As I wrote earlier, all members of that first championship team ultimately go to college, so the vision of Coach White ends up defeating that of Senor Valles.

Recall that all seven of these runners are Hispanic. And that reminds me of another issue that is in the news recently -- the accusation that Harvard admissions have an anti-Asian bias.

This falls under the umbrella of affirmative action. Note: I do not -- repeat, DO NOT -- speculate on the academic qualifications of the seven runners. I will not hurt them or their families by commenting on something that I don't know.

I do point out that Proposition 209, which banned affirmative action, was passed in 1996 -- nearly a decade after the film takes place.

Because I fear that writing more on this issue in this post might look like a speculation on the runners' specific academic records, I'll instead only post another link to an even longer debate thread between Alison Collins and Floyd Thursby. Again, what they write speaks for itself:

https://sfpsmom.com/so-whats-wrong-with-merit-based-enrollment/

Warning: There are many racial insults thrown around in the comments. The important thing I get here is that strong students are prepared to make a sacrifice in order to do well in school. And I believe that XC runners (regardless of race, as McFarland proved) are more prepared to make sacrifices in order to achieve success, whether it's on the XC course or in the classroom.

I will say something about the academic records of the runners in the classes I sub today. I tell the students that XC runners tend to be strong academically as well and ask the runners whether they are doing well in their classes. Three of the four runners say indeed they are.

But the fourth -- the girl in the fifth period class -- tells me that she currently has only a C-minus average this semester. Yet I see her carrying around a reading book for pleasure. Unfortunately, many students don't read for pleasure these days, and so when I see a reader, I figure that she's probably a hard-working student. I have no doubt that she'll be able to raise her grades by the end of the semester and I tell her so, because she's a distance runner who's used to working hard. She reminds me of my own XC teammate who went from being academically ineligible to run one year to becoming a team captain the following year.

And as for the non-runners, I tell them to think like a XC runner -- start out strong and cross the finish line hard. The next finish line for these students is winter break -- and they should work hard in their classes until they cross that finish line. Checking out early because there's only one day left, or one week left, until winter break is like easing up before crossing the finish line.

This is exactly what happened to a Chinese marathoner over the weekend (the day after the California State XC Meet, though due to time zones the races were mere hours apart). Someone throws a Chinese flag at her before she wins the race -- and as a result, she finishes second to an Ethiopian:

https://www.runnersworld.com/news/a25251849/chinese-runner-loses-marathon-flag-incident/

I really wish I can sub for this class one more day, so that I can watch the last part of the McFarland movie with these students and inspire them once more. I want to cross the finish line of the movie with them and watch the McFarland runners cross the finish line at the State Meet. But once again, I don't know whether I'll be called upon to sub on Monday.

Well, let's make sure that our students at least cross the Chapter 6 finish line. Here is the test:

Chapter 6 Review (Day 69)

Today is the second day of subbing in the high school Spanish class. As I mentioned yesterday, I won't do "A Day in the Life" today.

And that's with good reason too, because all Spanish I does today is watch a video. And you'll never guess what movie the regular teacher has me show the students today.

Do you give up yet? The movie I play today is McFarland USA -- that's right, the exact movie that I watched and discussed in my State Meet Saturday post!

This marks the second time this month that the students in a Spanish class I sub for watches a movie that I'd watched myself just days before subbing. The other was Pixar's Coco. The timing isn't merely a coincidence -- I watched Coco on Dia de Muertos (Day of the Dead), and the students viewed it a few days later. And I saw McFarland on State Meet Saturday -- and there's a good chance that this teacher is familiar with the date of the State Cross  Country Meet.

And both feature Hispanic main characters (the Mexican family in Coco and the Mexican-American runners in McFarland), which make them appropriate for a Spanish class. Today, the students watch McFarland in English with Spanish subtitles. The regular teacher could have simply had them watch the Spanish dub, but that might have been too tough for first-year Spanish speakers to understand.

In each class, there is at least one runner on this school's current XC team. I have that runner explain to what cross country is, including the exact distance (as this isn't mentioned in the movie). In sixth period the runner has a PR of just under 20 minutes, while in fourth period there are two runners, one slower and one faster than my PR of 18 minutes. The runner in fifth period is a girl with a best time of 32 minutes. I (jokingly) inform her that her time isn't good enough to make it to the State Meet (even in the girls' races, where the best times are around 18 minutes).

Each time I watch the film, I look at the scene where Coach White is researching times from previous races (not State Meets, of course, since this is the first one) and comparing the top times to the times of the McFarland runners. But even when I pause the film, the distance is nowhere to be mentioned.

I also notice the scene where Coach White notices that according to his car speedometer, his #1 runner Thomas Valles can run 12 mph. "That means you can run a mile in five minutes." But in the Spanish subtitles, these are converted to metric. Instead, Valles is described as running 20 km/h, or 1.5 km in five minutes. The distance 1.5 km (or 1500 meters) is a common metric equivalent to the mile run at the Olympics. In high school track, though, the 1600 is more common (and is actually closer to one mile, as I explained on Saturday).

Posted on a wall is a poster for the McFarland movie. Apparently, it's been autographed by both the actual Coach White and his wife. I'm told that a former student gave the regular teacher the poster.

In the Spanish II classes, the students don't watch McFarland. Instead, they're working on a packet that will review them for an upcoming test next week. They'll have several days to complete the review packet, since it's ten pages long.(How ironic is that -- two years ago today, at the old charter school, I sang "The Packet Song" about a ten-page packet, and today I see a real ten-page packet.)

But on the blog, though, today is the review day for our Chapter 6 Test -- and it's a simple two-sided worksheet, not a ten-page packet. That's why I'm posting it the day before the test.

This is what I wrote last year about this review worksheet:

OK, here is the Chapter 6 Review worksheet. I've made changes to the worksheet in recent years, including the activities I give the same week as the worksheet -- and due to these changes, I didn't have much to say about this worksheet last year. Since this week's activity was already given as a multi-day assignment the last two days, there is no activity to give today, so students should just ignore the instruction to perform an activity.

Activity: Corresponding Parts in Congruent Figures, Continued (Day 68)

Today I subbed in a high school Spanish class, at the same school as yesterday. It's the first day of a three-day subbing assignment, since this teacher will be out the rest of the week.

Should I do "A Day in the Life" today? That's a tough one. On one hand, a Spanish class isn't quite representative of what I want to teach someday, since I don't even speak Spanish. But on the other, it is a multi-day assignment, which means that I want to focus on my classroom management.

In the end, I decided that I will do "A Day in the Life" today, but not tomorrow or Friday. At least by doing so, today you'll be familiar with these Spanish classes. Then you'll be ready for my quick overview of my day in tomorrow's and Friday's posts.

7:55 -- Second period (recall that at high schools, "first period" is like zero period) is the first of the Spanish II classes. These students have three tasks -- first they must answer some questions out of the textbook, then they complete two pages in the workbook, and finally they have a worksheet.

8:50 -- Second period leaves and third period arrives. This is the other Spanish II class.

9:50 -- Third period leaves for snack.

10:00 -- It is now tutorial. One student is working on her Algebra I assignment, and so I am able to do some actual math today. Her assignment is on putting equations in standard form and graphing them using intercepts. I notice that the Glencoe text does something here that I wish other Algebra I texts would do -- admit that in the form Ax + By = C, A should be positive, and A, B, and C should all be integers with no common factor other than 1. Other texts don't state this assumption directly -- but  they make this unstated assumption when giving the answers at the back of the text. Then students get confused as to why their equation of -2x + (1/2)y = 3 doesn't match the answer in the book.

10:30 -- Tutorial leaves and fourth period arrives. This is the first Spanish I class. These students have a one-page lesson to read, and then their assignment is similar to that of Spanish II -- they do about 4-5 pages on a worksheet, two page in the workbook, and six questions in the textbook.

11:30 -- Fourth period leaves and fifth period arrives. This is the second Spanish I class. Some of these students were in yesterday's P.E. class. I let them know about yesterday's sub report (third period was the best behaved, sixth period the best walking the track, fourth period had issues). One student was in yesterday's fourth period P.E. -- fortunately he wasn't one of the troublemakers. He tells me that the P.E. teacher has scolded the five students whose names I listed. He wasn't sure exactly what their punishment was -- it might have been just a one- or two-point deduction, or it could have been a detention.

12:25 -- Fifth period leaves for lunch.

1:10 -- Sixth period arrives. This is the last Spanish I class.

2:00 -- Sixth period leaves. Officially, seventh period is the teacher's conference period. For the teacher, this means that he teaches five straight periods followed by a break. For me, it means that I get to go home early.

As you might expect, the more mature Spanish II classes are the hardest-working classes. I name third period to be the best class of the day, since this class is the most hard-working. Once again, there's a trade-off, as second period might have been slightly quieter. (Also, there are more tardies in third period than in second, a bit of a surprise.)

I also decide to name a best Spanish I class of the day. This is tricky -- fifth period is the quietest, but only because this class is much smaller than the other two first-year classes. In fact, fifth period becomes even smaller today, as one student transfers from fifth to fourth period. He is shocked when he finds only two open seats for him to choose from in his new class (as opposed to his old class, where there are two open rows).

But here's the problem I have with both Periods 4-5 today. As I write down names for my good list, I notice that most of the hard-working students in fourth period are guys -- while in fifth period, most of the diligent students are girls.

Therefore I name sixth period as the best Spanish I class of the day, since both guys and girls are listed among the hard-working. I hope that I can inspire the fourth period girls and fifth period guys to work harder tomorrow and Friday.

This is what I wrote last year about today's lesson:

Normally, I'd be posting today's worksheet, but today is just Day 2 of yesterday's activity. I feel guilty for making a school-year post without a worksheet. But then again, I felt even guiltier for never posting multi-day activities and thereby never giving students an opportunity to continue a worksheet before posting the next one.

So there's no worksheet for me to post today. The students should continue working on yesterday's "Corresponding Parts in Congruent Figures" assignment.

And now I feel even guiltier because I mentioned no Geometry at all in today's post -- the only math I wrote about today is Algebra I. Let me make up for it by at least linking to some Geometry. Here is a blog post I found by retired teacher Henri Picciotto on this week's Geometry topics -- glide reflections and congruence:

https://webcache.googleusercontent.com/search?q=cache:-MJhnBMrEjIJ:https://blog.mathedpage.org/2016/01/glide-reflection-and-symmetry.html+&cd=10&hl=en&ct=clnk&gl=us

I had to use the Google cache to find this post, so let me cut and paste it again here:

In my previous post, I introduced glide reflections, and explained their importance from the point of view of congruent figures: in the most general cases, given two congruent figures in the plane, one is the image of the other in a rotation or a glide reflection. (In some special cases, one is the image of the other in a translation or a reflection.) Another way to state the same thing, as commenter Paul Hartzer pointed out, is that if the composition of two of the well-known rigid transformations (rotation, translation, reflection) is not one of those three, then it is a glide reflection.

In this post, I will give an additional argument in defense of the glide reflection: its importance in analyzing symmetric figures. The Common Core does not have much to say about symmetry (see my analysis.) This is unfortunate, because symmetry provides us with connections to art and design, as well as to abstract algebra, and is very interesting to students.

Symmetry is deeply connected to rigid transformations, and can be defined in terms of those: a figure is symmetric if it is invariant under an isometry. (In other words, if it is its own image in an isometry.) In the most familiar example, bilateral symmetry, the isometry in question is a line reflection. Another well-known symmetry is rotational symmetry. In these examples (from my Geometry Labs), the stick figure is its own reflection in the red line, and the recycling symbol is its own image in a 120° rotation around its center:

     

Therefore the stick figure is line symmetric, and the recycling symbol is rotationally symmetric. These are example of symmetries for finite figures. They are known as rosette symmetries.

But what if we have a figure that is its own image under a translation? That is the case for this infinite row (or frieze) of evenly spaced L's. It is its own image under a translation to the left or to the right by a whole number of spaces:

...L L L L L L L L L L L L...

A frieze can be thought of as an infinitely wide rectangle, with a repeating pattern. A symmetry group is the set of isometries that keep a figure invariant. As it turns out, there are only seven possible frieze symmetry groups. In the example above, translation is the only isometry that keeps the group unchanged. But look at this one:

 

It is invariant under the composition of a horizontal translation and a reflection in a horizontal mirror. In other words, a glide reflection. If you want to analyze frieze symmetry, the glide reflection is absolutely necessary. 

Likewise, if you want to analyze wallpaper symmetry. In this Escher design, for example, the light-colored birds are images of the dark ones in a glide reflection (the reflection lines and translation vectors are vertical.)

Do all students need to know this? Probably not. But to some of us, this is a lot more interesting than many of the "real world" applications of math I have the opportunity to present.

--Henri