Wednesday, February 19, 2020

Lesson 11-4: The Midpoint Formula (Day 114)

Today I subbed in a continuation high school math class. Since this is actually a math class, I will do "A Day in the Life" today. Almost all of the classes today are Business Math -- either first semester or second semester.

8:00 -- First period arrives. This is a first semester class.

These students are learning about calculating property taxes. They begin by multiplying the total value of a property by a percentage to find the assessed property value, and then this by the tax rate to find the actual tax owed.

One traditionalist teacher, Darren Miller -- of "Right(-Wing) on the Left Coast," described a Financial Math course that might be similar to today's Business Math class:

https://rightontheleftcoast.blogspot.com/2020/01/the-advantage-of-teaching-financial-math.html

(Some of the comments are quite interesting, but I don't wish to make this a traditionalists' post.)

I have only one song that's somewhat related to Business Math -- "Compound Interest Rap." And so I perform it today.

Today is Fourday on the Eleven Calendar:

Resolution #4: We need to inflate the wheels of our bike.

I definitely stress this resolution today. Certainly, percents -- and how to convert them to decimal -- are worth including as part of our inflated bike-mind. When it comes to calculating tax rates, we also need to convert "per thousands" as well. The Business Math text refers to this as either "mils per dollar" or "dollars per $1000."

But the regular teacher also specifies a strict cell phone policy -- students have three chances to put the phone away, or I write down the name. She also specifies that anyone who talks back should be given a referral. This immediately brings in the fifth resolution on 1955 technology. Indeed, I mention the 1955 generation when reminding the students to follow this rule.

One girl refuses to put her phone away. When I told her about 1955, she repeats the phrase "Nice try!" and then ultimately the F-word. Since an F-bomb definitely qualifies as talking back, I end up writing the referral.

Even though I sing the song before telling the students about the strict phone policy, these students don't really enjoy the rap. Once again, this is understandable -- I've written before that sometimes older students (juniors and seniors) aren't as much into my incentives as younger students. Thus I don't always perform for such classes.

8:55 -- First period leaves and second period arrives. This is a second semester class.

These students are calculating the net proceeds from selling stocks. They must find the total value of the stock and subtract commissions and other expenses.

The fourth rule comes into play again, since one of the commission rates is 10%. Recently, some traditionalist -- I forget which website, though -- complained that many adults don't know how to calculate 10% of something. And so I tell these students that 10% (and 1%, of course) are two percentages which they should be able to find without a calculator.

On the other hand, the fifth resolution doesn't come up in this class. Second period is the best class of the day when it comes to working hard and following the phone rule.

9:55 -- Second period leaves and third period arrives. This is a first semester class.

As it happens, this is the worst class when it comes to following the phone rule. I must write down a total of four names -- three for listening to music and one for playing games and doing no work. Of course, the three students claim that they need the music to work in response to my claims that the fifty-fivers didn't need music to concentrate.

On the other hand, I don't need to write any referrals, since no one says the F-word to me.

10:55 -- Third period leaves for snack.

11:10 -- Fourth period arrives. This is a second semester class.

I only need to write down two names from this period -- but there's only four students (not counting the TA) present today, so that's half the class. One name each is for listening to music and playing games with no work.

But the only student who completes the assignment -- a huge disappointment compared to second period -- is the one who listens to music. This gives me an opportunity to balance out the negative with something positive -- I mention to the regular teacher that this guy completes the assignment as a mitigating factor. Perhaps she won't go as hard on him as she will the third period troublemakers.

12:05 -- Fourth period leaves and fifth period arrives.

This class isn't a math class. Instead, these students work independently on health -- or certain other classes -- via Edgenuity. (You might remember the online curriculum Edgenuity from that Summer 2018 fiasco where I'm chosen to teach a summer school class, only to have the class cancelled due to low enrollment.)

Once again, I'm directed to enforce the phone rule in this period. I must write down the name of one guy who listens to music on a phone.

This class also has a TA -- and surprise, surprise, it's the girl from first period to whom I've already given a referral! Then again, this gives me a second opportunity to turn a negative into a positive. I write down for the teacher that she ends up helping another girl start on her assignment. She does take out her phone again, but notice that this is OK since she's a TA -- and the regular teacher's own words are no phones "while working." She directs the fourth period TA to correct papers from earlier periods but leaves no work for this TA, and so this TA technically has no assigned work.

Notice that of course she doesn't help much -- and really, she can't, since the entire assignment is on the Edgenuity website. But little things matter -- and after I give this girl a referral in first period, I want to cheer her on after she does something positive, no matter how small. Even though I must tell the teacher what happens in first period, I'm hoping that the teacher goes easy on her TA, especially considering that she already received punishment for the morning F-bomb. If I ever return to my own classroom, I want to continue recognizing when punished students turn around and do some good.

1:05 -- Fifth period ends, thus completing my day at the continuation school.

And as continue to think about this student, I ask myself, is there anything I could have done to avoid the F-word and subsequent referral? I wonder whether I speak too loudly when I tell her to put the phone away. If I were more subtle, even if she refuses to put her phone away, she might have at least avoided cussing me out, and then I would have just written her name for the teacher as opposed to on a referral.

The fact that I must enforce the phone rule also gets me thinking about how what phone rules I might use in my own classroom. I already explained how I might designate certain days as Technology 1955, 1973, 1991, and 2020 depending on what technology they may use that day.

Anyway, notice how the third period students claim that they concentrate better with music -- and that fourth period is evidence in their favor as the lone student who completes the assignment is the one who breaks the phone rule and listens to music. Of course, this is a very small sample size (a class of just four students), but it's something to keep in mind. It might turn be worth it to let a day when the students can work independently without needing to listen to instruction from me (such as the review day before a test) be Technology 2020 so that they can listen to music. If it turns out that the students aren't really working on the review, then Tech 2020 is cancelled for the next test, and I return to having no phones on review days.

Lesson 11-4 of the U of Chicago text is called "The Midpoint Formula." In the modern Third Edition of the text, the midpoint formula appears in Lesson 11-7.

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

Lesson 11-4 of the U of Chicago text covers the other important formula of coordinate geometry -- the Midpoint Formula. As the text states, this is one of the more difficult theorems to prove.

In fact, the way we prove the Midpoint Formula is to use the Distance Formula to prove that, if M is the proposed midpoint of PQ, then both PM and MQ are equal to half of PQ. The rest of the proof is just messy algebra to find the three distances. The U of Chicago proof uses slope to prove that actually lies on PQ. Since we don't cover slope until next week, instead I just use the Distance Formula again, to show that PM + MQ = PQ, so that M is between P and Q. The algebraic manipulation here is one that's not usually used -- notice that instead of taking out the four in the square root of 4x^2 to get 2x (as is done in the last exercise, the review question), but instead we take the 2 backwards inside the radical to get 4, and then distribute that 4 so that it cancels the 2 squared in the denominator.



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