Unit Plan. Math 8: Fractions

http://docs.google.com/Doc?docid=0AVXQhrx0fazLZGR3bmJxd2RfNTRncGZ2ZndjOQ&hl=en

Since it doesn't format as-is on blogger, here's a google doc of my unit plan. enjoy!

Cycling Digits: An incomplete solution.

Short Practicum. Story!

I was asked to teach a lesson for October 28, 2009. That lesson was the quadratic formula. I was asked to teach this lesson twice in a row. Once in the period before lunch, and another the period after lunch so that I would be able to figure out what went wrong in the first period and fix it for the second period. This is the story of the first period: Block C.

In the days leading to d-day, I was busy writing out the lesson plan from which I would impart my knowledge to the students of the class. I would start off the class reviewing material from the previous day, hand out a quiz, and proceed with the lesson. This lesson consisted of presenting the quadratic formula, working through some examples and proving why the quadratic formula looks the way that it is. I had originally planned to prove the equation before moving on to do examples because that was the way I was taught the formula. Under advice from my sponsor teacher, and experience team teaching the students, I decided to take the advice and adapt my lesson plan.

On the day of the lesson, I was nervous. My palms were cold, sweaty and I had butterflies in my stomach. The bell rings to mark the end of the previous period. "Fifteen minutes", I said to myself as I packed my bag to go to the classroom across the hall from the student prep room. I opened the door. I crossed the hallway. I dropped my bag off to the side of the room. I unpacked my notes and readied myself, high on adrenaline. My sponsor teacher walks up to me and asks, "how are you feeling?". I replied, "I'm nervous". He offers me tea to calm me down, and I take him up on his offer. It helped. I review my notes, and the bell rings. D-day.

"Good morning!" The lesson began. Review questions were written up. Review questions were answered. The quiz was handed out. The quiz was answered. It was the right difficulty and length. The lesson was going well.

"Who thinks completing the square is tedious and painful to use?" Hands rose. "Here's a faster way of solving quadratic equations." Time to teach the quadratic formula. The quadratic formula was presented. Everyone was happy. Engaged to boot. Examples went up on the board. Lots of audience participation. Hands were freely rising to give the answers. The lesson was going well.

"Who wants to see why the quadratic formula looks the way it looks?" A few inquisitive looks. Not bad. I can proceed. I proceeded. I erred. Twice. I gave up the proof. The lesson wasn't going well anymore.

"Ten minutes" I thought to myself. "I've lost the kids. Time for a story." So I start off the story with "Did you know that. . ." The lesson finished.

So what did I learn? I learned that it doesn't take much to lose a class and one needs a back up plan to be able to bring the students back from a disengaged state.

Dividing by Zero, a Reflection

What do I think about the free write? The free write, I thought was a little loaded. Discussing division and zero separate from each other sort of avoids the idea of dividing by zero. I did manage to use some of what I wrote on the free write in my poem but except for a few lines, I managed to keep division by zero out of the picture. In fact, I think that I didn't manage to glean anything I didn't know before about division by zero from the free write. Maybe it's because that I've heard the arguments for why one cannot divide by zero. I think this exercise would work *with* a classroom that doesn't genuinely know why dividing by zero is so taboo.

What do I think about the poem? I think it'd be awesome to use this exercise in a high school class, paired with the free write. I think each student would be able to pull out some sense of why division by zero is not discussed from their own experiences with division and zero. It most likely would be fun and cool all at the same time. As a pedagogical exercise, I think that I didn't take away all that much from it apart from the idea that it could work in a classroom. I feel that I could probably explain this mathematical no-no to a variety of secondary school kids of all levels.

Dividing by Zero, The Free Write

Divide:

What does it mean to divide? How do you divide? What *can* be divided? Let's divide by 7. 7's a number that's very hard to divide with in the sense that you cannot easily tell if a number is divisible by 7. Other numbers like 2, 3 and 5 have mnemonics that you can easily tell. They're quick and painless to tell. 7 is much harder. I've gone too far from dividing. Does dividing mean counting how many somethings fit into a bigger something? I think that makes sense in the realm of integers. For non integers though it gets a little more weird. It's all well and done so I don't really know. I dont know what to write. I dont know what to write. Oh. Let's divide. a over b. b fits in x many times into a perfectly. and it's only x. it's not any number. it's cool like that.

Zero:

Zero. nothing. nada. zilch. Nothingness. Emptiness. like the amount of what i want to write. zero is kind of cool. when you multiply it to something, it reduces that something to zero. adding zero to something does not change that something. it's great for problem solving sometimes. but when you divide. what happens when you divide. zero over something is always and will be zero but what happens when you divide by zero? thats sort of a grey area in math. it's weird. it's like a black hole. a singularity. a void, even. a singular point. it's crazy. but we'll talk more about that later. let's talk about something raised to zero. something raised to zero is always 1. except zero raising itself to zero. it's one of those weird divide by zero things. crazy.

Dividing by Zero, a Poem.

How can you divide by zero

I want to know

Why would you want to divide by zero

I want to know

Is it practical is it trivial

I want to know

Is it fun is it boring

I want to know

How many zeros can fit

Too many says my head

Too little says my head

Just enough says my head

So many answers

So many suggestions

All numbers fit in my head

Is it zero is it one or is it fifteen-million three hundred-thousand two hundred and ninety-four

I do not know

I do not know

I want to know

I want to know

All I can say is that it cannot be any

Because if it is any then it is all

All are zero all are one all or all are fifteen-million three hundred-thousand two hundred and ninety-four

I want to know

I want to know

But I cannot know

But I cannot know

Team Teaching: Factoring Quadratics

THE GOOD:

My peers felt that the use of algebra tiles as manipulatives worked very well in explaining what the factored and expanded form of a quadratic looks like. It kept them very engaged and it was sort of a fun puzzle solving activity because finding the right configuration to get a rectangle was fun. It also helped students realized that there are some configurations that don't really work. As well, they found the lesson to be clear, concise and got the point across well enough.

I felt very much the same about what my peers felt. When Enrique, Nathan, and I were discussing how we would introduce factoring quadratics as such and got to discussing the algebra tiles, I saw that the activity would really engage my peers. I think this would totally work in a classroom one day. In fact, during my practicum, if I'm asked to do an introduction to factoring quadratics like this, I might adapt this lesson plan.

THE NOT-SO-GOOD:

My peers thought that time management was an issue. They also saw that there were some issues with classroom management as some students were chatting and not paying attention after the group activity had ended and some students felt that the group activity should have deserved a bit more time.

I saw the issue with time management as well. We tried our best to get our point across as quickly as possible but I felt that this lesson could have gone a full hour given the activities and the discussions that it could have garnered. I wanted to let my peers figure out the pattern between the two ways of calculating area of the rectangles they formed as this would show them, not only FOIL, but how to factor a quadratic polynomial. But as such, time was an issue and I felt that the point of the exercise was lost due to time constraints.

With regards to classroom management, I think I could have handled myself a little better. There were a couple of times where I "snapped" and shot the infamous "look" but I recovered quite nicely once I got the students' attention again. That might be something I want to look out for in my future practice.