## Thursday, 6 March 2014

### The Rational Divide

If you know me, you know I'm fond of "Day in the Life" initiatives. This week, there was a proposal to do a "Single Class Edition". (You can read about it on Tina's blog here.) That's new! In most of my previous ventures, I said "then I had this class" with minimal elaboration.

Quick background info: The course is 3U (Ontario Gr 11 Functions), the unit we're in is "Equivalent Equations" and we're currently simplifying "Rational Expressions". Introduced the topic Monday, concept of holes Tuesday, adding and subtracting Wednesday. Which brings us to Division/Multiplication.

There's 28 students in my classes. The desks are in three rows of 10, smushed together so students can talk with each other. I'll be breaking down a 75 minute period, 8:40-9:55.

#### GETTING RATIONAL

8:40 - Period begins. Morning announcements.
8:45 - Period actually begins. I have up the last slide from the previous day, where I rushed a bit to have ten minutes to hand back tests. Also hand back tests for a couple who were away. Start of class is time for individual questions based on homework. One student asks for the second last slide from previous day; I oblige, someone else wanted it too. Not getting much in the way of questions. Take attendance; students help when I realize that (once again) there's an absence where I'm not cluing in who, due to not seeing the face.

 First Slide
8:55 - New topic: Put up the multiplying/dividing slide, as well as homework for tonight (to start, for those getting it fast). Correct spelling of multiplication, thanks to a student. Give them some time to look at it and copy down, any other possible questions individually. One such question is student asking if answer is 1.5/8, I note answer is just 1.5

9:00 - Ask class how we divide those fractions. Student I was talking to answers: we find a common denominator, then divide the top. This throws me off because, well, I'm not used to getting that answer. I write up effectively that method, ask if anyone did other things. Another student says they flipped the second fraction and multiplied. I said yes, that works too, show that, and point out that the REASON it works is because, by dividing with the common denominator, things would become "unity" anyway.

9:05 - A student asks "so why did you even show us that?" Glad you asked! Because now we have to apply this, not to whole numbers, but functions. I walk through the first example of that, pointing out how the common denominator generates a domain restriction on x that wouldn't be visible otherwise. Also show how it can be done using the reciprocal of the second, but you STILL need to track the original asymptote. Check for questions. Someone says "back up!", I back up and reiterate a few things; again, homework's on the board for those already getting this.

9:15 - On to second slide with second example. Give them time to try that. Wander around to help individually. Couple people canceling around the additions, I remind we can't do that. Some having issues factoring the common denominator part. I keep wandering, students also talking with neighbours for help.

 Second Slide (after writing)
9:25 - I show how it factors, encourage keep going through the rest. One student notes there's a lot of "holes" for this graph. I agree, saying something like "Holey Question, Batman", get a few chuckles. Some are asking me if answer is correct - I honestly don't recall (made up these examples last year), so follow their logic, looks good. Now that I know, makes checking later ones easier.

9:35 - I put up the solution in full. I use the "common denominator" method, emphasizing that the reciprocal method works (careful with restrictions) and that canceling the (x-3) right off the bat works too "but I'll pretend I'm so zoned on Common Denominators that I don't notice - hey, I'll get the same answer". Check for questions. Guy is wondering what the result (x/(x+1)) actually looks like. I pull up internet and plug it into Desmos. Shape reminds me that, yeah, it's basically 1/x, moved one left, then vertically stretched by "x", so I say that. Someone who wasn't there the first day we did holes (when I zoomed in on the Desmos graph to show how it breaks) wants to see it zoomed. I start doing that, before remembering I didn't enter an equivalent equation - there's no restrictions here, as in the original equation. Derp.

 Third Slide. Too cruel?
9:41 - I put up my multiplication example, saying I'll see if I can backtrack to Desmos while they're working on this one. End up having issues because I'm already so zoomed in; a student next to the curious one has his laptop, says he'll just show her the Desmos whackyness.

9:46 - Conscious of time, I show the factoring for the multiplication example. Solve it. Ask if there's questions for the last five minutes, either about this or the homework - and yeah, look at those questions if you haven't yet. Student is wondering about the restrictions on last example, what with the fractions. I add the explanation to the slide, she asks "will that be on the website", I assure her it will be. Also show how monomials divide (without mentioning power laws) since I didn't get to my last example of such.

9:52 - Guy with the laptop shows me how when he moves the window a bit, the insanely zoomed in Desmos graph tries to redraw itself differently. Hadn't realized that before. Interesting.

9:54 - In the last minute, I show a couple of crazy graphs "like you might end up seeing in the Advanced Functions" course. Then, before the bell, remind them that there will end up being a quiz on this week... after the March Break. To see what's been retained.
Student: "I'll study." His neighbour: "You liar." Student: "I'm not lying!"
9:55 - Bell rings, class switchover.

#### BONUS: SECOND VERSE

Second verse, same as the first? Well, same material, different students, so not quite the same as the first period. But I'm thrilled to have the same course twice for the first time in a year and a half... so here we go. Spot the changes.

10:00 - Bell rings, I presume, I'm still talking to a couple students after having handed back yesterday's tests from those absent. Head to front to pull up same first division slide from before. (Forget to fix the spelling.) Remind again, first ten minutes or so for questions, if none, have a look at this. Back to the students; one offers the help the other, other says he doesn't like having to ask for help. I say something like "yeah, it's tough, but once you understand you can always help the next person, like the circle of life or something". First guy says it's the "circle of math", I say I like that better.

10:05 - Attendance. I keep circulating in the room. One question about previous day. One question about TODAY'S homework from a person who's already started it. Field those.

 Seen the Canceling Song Parody yet?
10:10 - Back to the front. How do we divide fractions?? This time I get the answer "you take the reciprocal of the second fraction". (Yes, student used the term reciprocal.) I respond with yes, but WHY? No one offering up the common denominator this time, so I show why doing that works, and why that means taking the reciprocal is valid. Some nods.

10:15 - I say apply one of those methods to first function example; don't walk through it immediately. Circulate. Some ask me if they're getting it; most missing the common factor of 4 but have the idea. I then put it up on the board, writing something more or less like in first period. I don't stroke through the denominator this time, more underline it; I like that better.

10:26 - Second slide/example goes up to try. More circulating just like earlier, and prior period. Some questions about this - some are about the factoring, others are getting it, but then I have to remind them to go back and deal with restrictions. The one girl from the start of the period has another question related to the new homework (it's a common factor issue).

10:40 - I've now put up the factored version of the question, as I think most people made it at least there. Mention the answer (sans restrictions), encourage keep trying it.

10:45 - Full solution now on board, I offhandedly make a "Holey Batman" comment again. (It worked better first period.) Mention how the simplified graph relates to 1/x, say that I showed it this morning, do people want to see it? Some nods, so pull up Desmos again, there we go. Someone wonders how that came from the original division. I say I'll pull that up, but first, multiplication question to try.

 Why you gotta be So Mean?
10:48 - Multiplication question up, I note how the operation is generally easier, as factoring makes everything multiplications anyway... so the factoring for my question is quite difficult, to keep things interesting. Student: "That's not nice." Me: "No. Sometimes life is cruel."

10:55 - I show in Desmos how the two graphs collide to make the division from previous example. Question about the nature of holes (and why it's hard to show them on a graph), I note how a value works for 3.00000001 and 2.99999999 but not 3 - how to show that on a zooming scale? Keep circulating. More factoring issues. One student in particular seems to be tuned out, but I see he's made some progress today, so I'm not going to push it. I remind everyone that I do pick tricky examples to show at the front, do have a look at the homework, might be more straightforward.

11:05 - I put the solution on the board, making sure to show tile chart for the hardest one, as I did in first period. Student asks to see chart for the cubic. I do, but point out that as it's just a common factor, there's just the one row. Pause. Me: "Did that help?" Student: "No." Honesty is good. I head around to talk with her and her neighbour.

11:10 - Back up at the front, quickly show the same monomial division as I did in first period, in case people need it. Also show the few crazy advanced functions style graphs, and remind of quiz. I mistime things - pretty much done talking with 30 seconds left. (Lost my watch yesterday, actually. Found it when I got home today.) So students start shuffling to the door. Blah. Pet peeve, but I quip about how people seem eager to get to lunch.

11:15 - Bell goes, lunch starts. One student had talked about getting help at lunch, but either something I said individually answered the problem, or he forgot. There's another student who'd asked to talk, slipped my mind to ask if lunch was possible until too late, hopefully we connect up again.

#### ROUNDING THINGS OUT

No time for a lunch break for me at this point! Anime Club meeting in my room; ended up watching Lucky Star. Third period after lunch was my prep, had to deal with quizzes and math competitions (though more made notes for this post). Fourth period - data management (4U statistics)... the second unit of which I've been messing up, after doing a decent first unit. Sang "It's Probability", Carly Rae Jepsen song to close off the period, which I'm waffling on adding to my published material.

After school, I'm spearheading the musical rehearsal until after 5pm, so not leaving school until 6pm at the earliest. Coincidentally, 6pm is around when I made it to the washroom for the first time - probably helps that I never got around to drinking anything since leaving the house in the morning. Despite the fact that I still have marking to do, I wrote this thing up and came home.

I feel like I do more "direct instruction" teaching than not. Random activities isn't my thing. So I try to balance "me slapping down examples" with letting the students talk more. Some days I succeed at that better than others. Questions? Advice? Let me know in the comments. Hope you enjoyed reading.