ONE
On Thursday, I’m going to be introducing absolute value inequalities. Last year I used the picture below as motivation.
I then tried to work backwards to show kids absolute value inequalities. It wasn’t too hot a success. Certainly the “application” wasn’t a motivator, and working backwards just confused things.
This year, I’ve decided to start with a warm up. Without them knowing anything, I’m going to ask them to do this for the first 7 minutes of class with their partners.
I already can see the great questioning and discussion that this simple worksheet will generate between partners. And then, when we come together: WHAM! powerful! It’s a simple thing, but Oh! So! Delicious!
After that, after we see some patterns and make some conclusions… then, then I can throw up the picture of the bag, and talk about it meaningfully. And have kids work backwards from their own conclusions to finding a way to express that region mathematically, using absolute value inequalities.
TWO
I’m introducing limits tomorrow. I pretty much have carte blanche in what I do. Last year what I did was sad. Like SAD. Like: “Here’s what a limit is. Get it?” This year, I’m stealing pretty much from CalcDave wholesale. Here’s his calculus questionairre. And here’s what I made.
Pretty much the same thing. Then I’d like to somehow have them start thinking about how to get velocity from a position versus time graph. Haven’t quite figured that out yet. Either that, or Zeno.
Continuous Everywhere but Differentiable Nowhere 
Good job including #2, 3, 5 on the worksheet. How have you defined absolute value? I have found that defining |x-y| as the distance from x to y on the number line supports equation and inequality solving well, helps students find “the second answer” when there is one, helps students see why a statement like |x-4| < -5 is an absurdity, and supports notions of absolute value and limit in higher dimensions. (For example, the notation |x-y| would have the same definition in taxicab geometry… in complex numbers… in any dimension, really.) Best wishes and good luck!
Interesting post – thanks for continuing to make your learning public for us all!
I found myself drawn to the last two ?s of the Limits Prelude, and will be curious to hear how the kids respond to the task itself (seeking out the graphs) and also their comparison between the graphs that each of them finds (as they will surely be so different, and yet so alike in that one small window).
As far as reading velocity from position vs time graphs, my colleague Tony Wayne and I used to have our students try out a couple of different activities in our physics classes:
– We used this animation (http://bit.ly/dnjA4F) and its corresponding worksheet guide in a large-class setting (with the animation projected – be sure to turn on the speakers, too). It is sort of a guided inquiry lesson. The teacher would pass out the worksheet guide associated with these graphs and then present the graphs individually to the class. The first 4 animations are associated with the motion of an insect in one dimension. Students would watch the bug move over a given time, and then try to draw the graph that describes the motion. (They would probably have to see the motion several times.) In the lesson, the teacher would use A LOT of think-pair-share – we would discuss every graph together as a class, having students share their ideas and questions before going on to the next graph. Pressing the “curve” button shows the graph that is associated with the motion. In Graph 5 and 6, the students see the graph first, and try to describe the motion of the bug in words. The goal is develop the students’ visualization skills, and for them to make the connection between slope on a position vs time graph and velocity.
– If you’ve got probeware that allows for a motion sensor, there’s a great “Graph Match” activity follow-up where students try to match the motion described by a given graph. Engages the students, and seems to stick with them. (As an added bonus, you can change the graph settings to be a velocity vs time as opposed to position vs time…you can imagine the students’ realizations when they see the “same graph” describe such different motions.) Something I’d be happy to look for if you’ve got probes available.
Can’t wait to hear how the students respond to the activities you wrote about – keep on keeping on!
AAAH! That’s such an awesome resource. Thank you! I immediately showed it to the calculus teacher next to me, and then another math teacher at my school screamed: she student taught at Albemarle High School! She asked me to ask you: “Do you know Carla Hunt?”
Glad to hear that the resource is helpful! Tony (the creator of the animation) has a resource page for teachers with other materials: http://www.mrwaynesclass.com/teacher/index.html
(Of course, as you continue to post your ideas and challenges, I’ll be sure to keep responding here with questions and comments.
And yes, I DO know Carla Hunt. Quite well, actually- she and I were both at AHS, and had since both embarked into new roles within our county’s instructional coaching program. That’s so wild that your “neighbor” worked with her- I’ll be seeing her tomorrow, so I’ll let her know about the connection!