Monday, June 20, 2011

Day 4

As a reminder, my Algebra class meets four days a week (MTWF), so day four is going to be a Friday but it will be two days after our day three meeting on Wednesday. Today is really the first day that is my "own" in Algebra. The first semester of Algebra at my school is basically all about linear equations, and I start by taking a look at proportional thinking - more specifically, ratios and proportions.

My opener today is designed to build off the skills review/assessment they just had and help front-load some ideas for today's lesson. At the moment (assume that phrase for all planning posts on this blog), this is what it looks like:

As a reminder, students are expected to begin the opener by the time the bell rings. They then work on it for a little while individually, then discuss with their group members, then students come up to the smart board and explain them to the class (with that getting saved to PDF and posted to the class blog).

We then open the lesson with a bit of vocabulary.

Then I try to lead them into solving proportions by starting with something they hopefully can do somewhat intuitively: "When you divide some number N by 2 you get 12. What's the value of N?"

From that we move slowly toward doing the inverse operation, to "undo" dividing by 2 we would multiply by 2; to undo dividing by 4 we would multiply by 4.

But then what happens when we throw a fraction into the mix? Hopefully they'll see that the same principle applies. From there they work through several examples with the variable in the numerator, then we try to extend to having the variable in the denominator.

At this point I really want them to focus on inverse operations, so I'm not showing them to "cross-multiply" to solve proportions.

Assuming things are going reasonably well at this point, I then introduce a hopefully somewhat interesting application of proportions.

In my head we still have about 10-15 minutes left in class (but often my head is very, very wrong), so we'll try to make a quick geometry connection. (If we don't have the time, we'll skip this - perhaps taking a look at it the next day.)

How ever much we get through gets posted to the class blog as a PDF.

We will then talk briefly about how I expect them to work through (and take notes on) the videos, as their homework will be to watch/work through the Solving Proportions and Percents video.

While I'll probably tweak it, this is what I wrote last year on the blog to follow-up what we talked about in class.
We previewed the Solving Proportions and Percents video that you're going to watch for homework tonight and talked about the different pieces in it, how you should use it, and what you need to write down in your notebook. There are three main parts to the video: an Examples and Explanation part, a Guided Practice part, and a Self-Check part.

Examples and Explanation: Just what it sounds like. I explain how to do the problems and work through some examples. You don't need to write anything down (unless you want to), just watch, listen and learn. Pause the video and replay parts if you need to.

Guided Practice: I give you a problem, then ask you a series of questions with about 5 second pauses between questions for you to think about it and answer it for yourself. If you need to, pause the video to give yourself more time. Again, you don't have to write anything down here (although you can and it may be a good idea to).

Self-Check: I give you a problem, ask you to write it down in your notebook and solve it, then I show you the solution in the video. Once the problem is on the screen you need to pause the video, write it down and solve it, then play the video again to check your work. You may need to pause the video again to view the solution if you need more time. These problems you definitely need to write down in your notebook.

Remember, you can always replay any part of the video you need to go back over something.
I would love to hear your thoughts/feedback/suggestions for improvement on any/all of this.


  1. I like the progresssion of skills here and I have a feeling you are right about not having enough time to jump to Geometry (then again, your class might just surprise you).

    Here are a couple of other approaches I've tried (links to posts):
    1. Problems vs Exercises inspired by @dcox21 which, in terms of your lesson above, start off with the dust mite problem

    2. Teaching Equations Big Picture style and its corollary Algebraic Equations Virtual and Real which contains links to some sites for online drill (supplement your video perhaps?)

    Like you, I allocated time on vocabulary. I'm big on literacy in maths - there is a reason why most terms have been used. In your lesson above, one of the things I would emphasise is the notion of inverse being opposite. Most students learn the four operations as opposite of each other (+ with -, * with /).

    Like you, I'm averse to teaching cross-multiplication. It's a shortcut that is meaningless until they understand the principles that allow it: equality and inverses. When I saw a student doing this - probably tutored - I asked her why it worked to make sure she understood (she didn't).

    Can I just say you are so thorough?


  2. Hey Karl,

    Just a couple of quick ideas. I'd love to talk in skype sometime if you like.

    In the warm-up, Q2, I'd ask "How many different ways can you write the number 4?" This can lead to good discussions about the meaning of the equal-sign and multiple representations of numbers which is essentially what a proportion is; 2 ways to represent the same number, or ratio. This flexibility in thinking pays off huge dividends down the road.

    I like the motivation for starting off with the mite problem but I'd actually start with the AHS gym problem, it's closer to their personal experience; 1st picture/question only. Let them wrestle with it a bit; see what ideas they come up with. Too much scaffolding too soon in the second picture (although maybe I'm assuming too much about how quickly you planned to move from one to the other).

    I'd move to the mite problem next, but I would end it at: "... it can lift 0.000253 lbs." Then ask, what questions might we ask about these mites? If they struggle, maybe ask: "What would a superhero called MiteMan be able to do?"

    I'm also a big fan of teaching proper mathematical vocabulary but maybe wait until the need for each term emerges out of the lesson:

    "OK folks, it's taking us too many words to say 'two different ways that are really the same to write the same number written as a fraction'. We've got a word for that in math; anyone know what it is? Do you wanna know or should we keep saying it this way? ... You really wanna know? OK, it's called 'A Proportion.'"

    In this way they learn we've got good reasons for speaking funny in math. ;-)


  3. Malyn and Darren - Thanks so much for your thoughtful comments – this is exactly the kind of feedback I’m hoping to get.

    What you both suggested are things I considered and struggled with as I was planning the lesson. Philosophically, I want to start with the problem (whether the dust mite or the gym problem) and let them struggle with it. My concern is, though, that in my experience they don’t ever get there. Specifically with the gym problem, I don’t think it’s realistic they are going to come up with the idea of using similar triangles and proportions to solve it. (Darren, I do allow a little bit of time before the second picture, but probably not enough.) That’s why I chose to lead them through at least a somewhat inquiry-based approach to how we would solve a proportion, and then front-load the gym problem with the idea of similar triangles having proportional sides.

    And while I value allowing them to explore the problem and perhaps come up with other ways (e.g., measure the height of the cinderblocks and count how many there are), this is where I run into the time/curriculum issue. I have to teach them proportions and only have so many minutes with them, so I can’t “allow” them to get off track too much. That frustrates me tremendously, but ends up in the compromise you see. So I’m considering starting with the gym problem, but only just showing them the problem and then telling them we are going to learn some things today that will help us solve it. That at least embeds today’s work in something semi-real-world.

    I also spent a lot of time going back and forth on whether to start with the vocabulary or only bring it up in the context of the problems, and chose to frontload it for time purposes, but I’ll rethink that again.

    Darren, I love your suggestion on making more explicit “how many ways can we write the number 4” – definitely going to use that.

    Malyn, thanks so much for the links, I need to find some time to explore your blog. I’m hoping your “thorough” comment was a compliment . . . I’ve decided to take it that way :-)

  4. Hi Karl...

    Hmmm...? A tropical mite???! What about changing it to a bed bug. Bed bug may be more relevant to a 9th grader in terms of gross-out factor ;-) Two proportion problems, size of (bed bug:9th grader); and (time to fill up a bed bug: time to finish 711 BigGulp)? What'da think? :-) ...fwiw, I'm not a math teacher. My point being making algebra interesting to the uninterested student is tough. Maybe using strategies that will allow a student at their 10 year reunion to say, "..remember that crazy math teacher that taught us proportions with bed bugs..." If they can say that, you know you had them :-) Everything else looks solid!

  5. It's nice to see so many math teachers who believe strongly in proper math vocabulary! That said, I'm a little concerned that students might read "ratio" and "proportion" in the learning goal and improperly tie those to the skills in the review. Any chance the learning goal would work better if stated after the review?

    As for leading with the mite problem, I'd introduce it at the beginning (perhaps even before the review problems), but wait until later in the lesson to give students dedicated time to work on it. Just do enough to plant the seed in their head that the math they are about to learn is applicable and the problem will be solvable if they learn the skills and gain the understanding provided in the lesson.

    I like your use of the word "undo." I'm going to contradict myself a bit here, but I bent my "strict math vocab" rules by using terms like "unadd," or "unmultiply" before jumping to using the term inverse. It sounds like you have the same idea.

    On your slide with the triangles, you define them as similar because their angles are congruent. I'd assume students who don't know what similar means won't know what congruent means, either, so perhaps there's a better way to define that. Instead of words, what about providing a number of examples of pairs of congruent shapes, contrasted with pairs of non-congruent shapes? I know this is part of the lesson you're not sure you'll have time for, but these understandings are important and will pay off down the road.

  6. Raymond - I hadn't thought of the issue of the learning goal getting confused with the opener - I'll have to think about that more. For me, that was just a convenient place each day for them to have an idea of what we were going to be working on.

    I don't have a good feel for what the students know in terms of geometry and congruency, so I guess I'm just going to hope I run out of time :-)