In recent posts, I've been writing a COVID-97 What If? -- that is, what if the pandemic had started, not in December 2019, but in December 1997, when I was in high school? I would have lost most of my junior Track season in 1998, and started my senior Cross County season in February 1999. By May 1999, I would have moved on to my final Track season.
I created this COVID-97 What If? by taking my alma mater's current 2021 Track season and mapping it back to 1999. Last Saturday -- May 8th, 2021 -- my school participated in the Irvine Distance Carnival, and so let's say that I'd have participated in an invite on Saturday, May 8th, 1999 in COVID-97 world.
Notice that on the original timeline (that is, the actual year 1999, with the pandemic 20 years away), the week of May 3rd-8th was League Finals. That week, I ran in my final race, which was a 1600. For this What If? it was always my plan to run my actual 1600 time in whatever race I'm in that same week. On the original timeline, it was League Finals, but on the COVID-97 timeline, it's a Saturday race:
On May 8th, 1999 (a Distance Carnival), my 1600 time would have been 5:07.
Unfortunately, I'm not sure what invite I could have run in on May 8th, 1999. The Irvine Distance Carnival has only existed for ten years, so it wasn't around in 1999. In an earlier post, I mentioned the Covina Invitational, which did exist in 1999. But the real 2021 Covina Invite was cancelled -- so if we're truly mapping 2021 back to 1999, we'd have to cancel the 1999 meet as well. The only invitational this weekend that definitely existed in both 1999 and 2021 was Arcadia, but only the fastest runners get to run in it. My 5:07 time is nowhere near good enough for Arcadia.
Most likely, some school would have come up with a distance carnival for us, but there's no way for me, typing on a computer in 2021 in the real universe, to know who would have hosted it. For simplicity, we can assume that Irvine High School would have created an invite on the fly in 1999.
OK, so I have a location, Irvine, and a time, 5:07. It now remains to determine how this time would have compared to my teammates -- in both 2021 and extrapolating back to 1999.
In 2021, only two runners from my school run in the 1600 -- one junior and one senior. The senior's time was just under six minutes, but the junior was merely one second ahead of my own time. I'd like to say that I would have kicked harder in an effort to beat him, so that my final time would be 5:05.
In fact, I believe that my PR (or personal record) for the 1600 was 5:05, not 5:07 -- it's just that I ran that 5:05 a few weeks before League Finals. Moreover, 5:05 still isn't my fastest mile ever -- I believe that I ran 5:03 as the first mile of a three-mile XC race. (Here's what happened -- it was a Junior Varsity race, and I didn't want the opponent school to beat us, so I tried to keep up with their second runner. As the coach gave my one mile time, I knew I was running too fast, and slowed down for my second mile before recovering for the third mile. I ended up winning the race a full minute ahead of my opponents -- those guys trying to break five minutes for the first mile just barely finished three miles under 20. The moral of the story is run your own race -- don't worry about what the opponents are doing.)
Like many invites, Irvine has only Varsity and Frosh Soph levels, so any upperclassman must run in the Varsity race. Notice that the one senior who ran almost six minutes this weekend would ordinarily be nowhere near Varsity, save for the fact that our real Varsity runners contested the 3200 instead. Some of our other fast runners ran the 800 or even the 400 instead -- doubling the 400 with another event, including two racers who ran the 400/3200 double. (Also, I have no idea why "Distance Carnivals" would even have a 400 race. Since when is an event where runners stay in lanes the whole time -- and there's no bell lap because the entire race is a lap -- considered to be a "distance" event?)
Based on my memory, I believe that our 1999 Varsity runners would have done the same -- our fastest distance runners would have contested the 3200, leaving the 1600 open for runners like me. This is why I've been focusing on the 1600 in this What If? -- I'm opening the possibility that with the season extending past May 8th, I could keep improving my times and advance past League Finals into the next race, CIF Prelims.
Oh, and by the way, the State Meet for Track has already been cancelled. Of course, I know that there's no realistic way for me to make it to State -- the goal for this What If? is CIF Prelims.
This week, my alma mater is scheduled for one midweek meet and one weekend invitational, just like last week. But the midweek meet might be another nonleague meet -- and both races this week might focus more on relays. I'll discuss this more in my next scheduled Track post, on Thursday.
This is what I wrote two years about today's lesson:
Question 5 of the SBAC Practice Exam is on solution steps:
A student solved 3/(x - 4) = x/7 in six steps, as shown.
Step 1: 3 = x(x - 4)/7
Step 2: 21 = x(x - 4)
Step 3: 21 = x^2 - 4x
Step 4: 0 = x^2 - 4x - 21
Step 5: 0 = (x - 7)(x + 3)
Step 6: x = -3, x = 7
Which statement is an accurate interpretation of the student's work?
A) The student solved the equation correctly.
B) The student made an error in step 2.
C) The student made an error in step 5.
D) Only x = 7 is a solution to the original equation.
We notice that this is definitely a second semester Algebra I question, since we obtain a quadratic equation in Step 4. The problem is asking us to check the solution.
As far as I can tell, there are no errors in the solution. We can double-check the specific steps mentioned in the answer choices. Step 2 is correct -- instead of dividing by 7, we multiply both sides by 7 -- and Step 5 correctly factors the quadratic polynomial. We can also check whether -3 is an extraneous root or not:
3/(-3 - 4) = -3/7
3/-7 = -3/7
-3/7 = -3/7
Therefore both values of x are correct solutions, and so the answer is A).
The girl from the Pre-Calc class correctly answers A). But the guy from the Pre-Calc class answers B) and not A). I'm not sure what error he finds in step 2 -- the cross multiplying in that step is correct.
Question 6 of the SBAC Practice Exam is on parallel lines in Geometry:
When a transversal intersects a pair of parallel lines it will create two pairs of alternate exterior angles.
Ricky claims the angles within each pair are congruent to each other, but not congruent to either angle in the other pair.
Part A
Draw a transversal through the point that supports Ricky's claim, or select NONE if there is not a situation to support the claim.
Part B
Draw a transversal through the point that refutes Ricky's claim, or select NONE if there is not a situation to refute the claim.
Well, we finally reach a Geometry question. But this question is tricky for Ricky. We notice that this question mentions alternate exterior angles, which don't appear in the Second Edition of the U of Chicago text. They do, however, appear in Lesson 5-4 of the modern Third Edition of the text.
Let's assume that Ricky knows that alternate exterior angles are congruent. But notice that Ricky is making two claims -- one is about which angles are congruent and the other about which angles are not congruent. In Part A, the transversal must satisfy both claims, while in Part B, the transversal must fail at least one of the claims. Since Ricky's first claim is always true, any counterexample for Part B must fail his second claim.
A good question to ask is, how are the angles in each pair related to each other? Well, it's easy to see that one angle in each pair forms a linear pair with an angle in the other pair. Thus the angles in the other pair are supplementary. If a counterexample to Ricky's second claim exists, the angles must be both congruent and supplementary. Such angles are called "right angles."
Therefore the transversal for Part B must be perpendicular to the parallel lines. It follows that the transversal for Part A must be oblique to the parallel lines.
The girl from the Pre-Calc class correctly answers an oblique line for Part A and a perpendicular line for Part B. But her transversals don't go through the point that is given on one of the lines. The question clearly states "draw a transversal through the point," and so the SBAC might not give her credit for this answer. I know that the guy from the Pre-Calc class certainly won't receive any credit for this question -- because he just leaves this question blank. Hmm, this is the page on which both students start to struggle a little.
SBAC Practice Exam Question 5
Common Core Standard:
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
SBAC Practice Exam Question 6
Common Core Standard:
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Commentary: Solving quadratic equations by factoring appears in Lesson 12-8 of the U of Chicago Algebra I text, while proportions appear in Lesson 5-7 -- but notice that the steps above don't simply cross-multiply a proportion. Alternate interior angles appear in Lesson 5-6 of the U of Chicago Geometry text, but alternate exterior angles appear only in the modern Third Edition of the text. But students unfamiliar with alternate exterior angles might misread "exterior" as "interior" -- which would nonetheless lead them to the correct transversals for Parts A and B anyway. But even Pre-Calc students are having trouble with these questions, which suggests that students need to pay attention more to detail.
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