Today I subbed in a high school math class. It's in my LA County district. Since it's a math class, I'll definitely do "A Day in the Life" today.
8:30 -- Second period arrives. This is the first of two Pre-Calc classes.
These students are learning about polar coordinates and graphs. I've mentioned this on the blog -- there seems to be more emphasis on polar graphs now than when I was a young Pre-Calc student. I see some old colorful graphs on the walls -- these were created by students from the pre-pandemic year. It appears that while this might be interesting, this isn't a pandemic-friendly activity.
Choosing a song for this class is tricky -- I don't have very many songs for higher math (that is, Algebra II and above). I end up choosing "U-N-I-T Rate!" with the justification that some of my students are seniors who are a few months away from college hence the UCLA fight song parody. Then again, it's technically an honors Pre-Calc class, so there are many more juniors than seniors in this class.
9:40 -- Second period leaves for snack break.
9:55 -- Fourth period arrives. This is an Algebra I class.
Like many Algebra I kids, these students are currently solving quadratics. And I've also mentioned on the blog that as a young Algebra I student, the Quadratic Formula was the last lesson of the year. These students have a Chapter 9 Test coming up (before, presumably, finals week). Since the test is soon, I call on the in-person students to solve one equation from the practice test on the board.
This time, there's an obvious song for me to perform -- "Quadratic Weasel."
11:05 -- Fourth period leaves for second snack.
11:20 -- Sixth period arrives. This is the second of two Pre-Calc classes.
Both Pre-Calc classes have a large number of absent students. I wonder whether there's an AP exam going on -- obviously it wouldn't be the AP Calculus exam for these Pre-Calc kids, but it could be US History or another exam that juniors are more likely to take.
Because of the absences, there's only one in-person student (and a handful of opted-out kids). I tell her that she just happens to have the same first name as the lone in-person girl in one of my long-term subbing classes (the Math Skills class during my final week there).
12:30 -- Sixth period leaves. As it turns out, the regular teacher hasn't set up the Zoom for academic support, which means that I don't need to stay for it.
In the Pre-Calc classes, I remind the students that they must finish taking notes from the lesson before they can log off after their half-hour of Zoom.
Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:
Find x.
(This is another problem where all the givens are in an unlabeled diagram -- and it's a transversal problem, so it's especially tough to describe and visualize. One of these days, I need to come up with a better way to describe such problems. All that matters is that two lines are cut by a transversal, and one of the obtuse angles formed is 154 degrees and one of the acute angles formed is x. They aren't exactly same-side interior angles as one of them is exterior.)
As we approach finals week, all of the recent Rapoport problems plus our SBAC questions can make up our own finals review.
The important thing here is that x + 154 = 180, so x = 26. Therefore the desired angle is 26 degrees -- and of course, today's date is the 26th. (The diagram also contains an irrelevant 87-degree angle.)
Actually, this week is already finals week -- League Finals week, that is, for Track. The last day that leagues can hold finals is this Friday, but my alma mater's league has always held Prelims a few days earlier, on a Wednesday.
I attend my League Prelims today -- well, since this is still technically "A Day in the Life":
4:00 -- The 800's at League Prelims begin.
I try to identify certain runners, but it's tough because the announcer gives confusing information concerning how many heats there are in each race. In particular, he announces that there are three heats for the boys Frosh Soph 800, but the team member whose season we've been following on the blog (only because his times are similar to my PR's) appears to be in the third race (and he's a junior, so he can't be in the Frosh Soph race). And indeed, his time appears to be around our usual 2:20-something.
The winning time in the first few 800's (the Frosh Soph races) is around 2:12, with top times in the other races approaching 2:00. Once again, it's difficult for me to tell how many runners in each race advance to the League Finals on Friday due to the announcer's confusing info.
Returning to my COVID-97 What If? story, notice that I did say I'd focus on the 1600, not 800. I don't see any 1600 race today -- and if I recall from my actual season, the 1600 and 3200 are run only as finals, not prelims, and so the longest distance races won't be contested until Friday.
We'll continue discussing my COVID-97 Track season in Friday's post, since that's the day of the race.
Let's get back to SBAC Prep/Final Exam Prep.
Question 29 of the SBAC Practice Exam is on angles of elevation:
Emma is standing 10 feet away from the base of a tree and tries to measure the angle of elevation to the top. She is unable to get an accurate measurement, but determines that the angle of elevation is between 55 and 75 degrees.
Decide whether each value given in the table is a reasonable estimate for the tree height. Select Reasonable or Not Reasonable for each height.
Reasonable Not Reasonable
4.2 feet
14.7 feet
24.4 feet
33.9 feet
39.1 feet
58.7 feet
Once again, let's use h for the height again. Angle of elevation problems usually depend on the tangent ratio, where h is the height and 10 is the distance to the tree:
tan theta = h/10
But we don't know what the angle of elevation theta is, except that it's between 55 and 75. So let's try solving the problem for both of the extreme values:
tan 55 = h/10
h = 10 tan 55
h = 14.3 feet
tan 75 = h/10
h = 10 tan 75
h = 37.3 feet
And the true height of the tree can be anywhere in between. We thus choose Reasonable for Emma's three heights in this range -- 14.7, 24.4, and 33.9 feet -- and Not Reasonable for her other three -- 4.2, 39.1, and 58.7 feet.
Both the girl and the guy from the Pre-Calc class correctly answer this question. Both of them draw right triangles to help them. The girl writes her work for 55 degrees, but not 75 degrees, while the guy starts to use sine, then corrects himself to tangent. Most likely, both of them enter 10 tan 55 on their calculators, so it was obvious that they needed to enter 10 tan 75 without writing out the work.
Question 30 of the SBAC Practice Exam is on modeling with linear equations:
Emily has a gift certificate for $10 to use at an online store. She can purchase songs for $1 each or episodes of TV shows for $3 each. She wants to spend exactly $10.
Part A
Create an equation to show the relationship between the number of songs, x, Emily can purchase and the number of episodes of TV shows, y, she can purchase.
Part B
Use the Add Point tool to plot all possible combinations of songs and TV shows Emily can purchase.
Since each song is $1 and each episode is $3, it's clear that the equation is 1x + 3y = 10. Notice that I include the coefficient for x even though it is 1, because the SBAC interface for Part A requires a coefficient for both variables.
For Part B, there are four possible solutions -- (1, 3), (4, 2), (7, 1), and (10, 0). These solutions are discrete, but I suspect that the SBAC interface automatically connects the points to form a line -- the graph of the linear equation 1x + 3y = 10.
Both the girl and the guy from the Pre-Calc class correctly answer Part A. But the guy's graph isn't linear, because he miscounts and graphs (6, 1) instead of (7, 1). The girl's graph is linear. But both of them miss the solution (10, 0) -- which is valid, as Emily could have bought 10 songs and no shows.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Commentary: Both the tangent ratio and angles of elevation appear in Lesson 14-3 of the U of Chicago Geometry text. Meanwhile, Lesson 8-8 of the U of Chicago Algebra I text is called "Equations for All Lines." In that lesson, linear equations in standard form Ax + By = C are given, and it's stated that lines in standard form often arise naturally from real situations.
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