Desmos, Man!

For those of you teaching function transformations in Algebra 2 (or even Algebra 1?), check out this slick Desmos resource called Des-man that they put together recently with Dan Meyer and Christopher Danielson.  I know a number of teachers at my school who have done something similar to this over the years, even with the Desmos.com graphing calculator.  What's slick about this is that it provides a  nice short demo for kids to see how to constrain the domain and range, and it allows you as a teacher to set up a "class" and see all students projects in real time.  Very cool!

Another Desmos-made activity I found this summer is an activity called "Penny Circle"  that uses best fit sliders to find a function that optimizes number of pennies fitting into a circle of given radius.  It works out to be quadratic, I believe.  Again, a very slick and kid friendly/intuitive activity.  I'm going to do this with students in my quadratics chapter this year.

I'm VERY impressed with what's coming from Desmos these days.  Free online graphing calculator (that's superior to TI-84s, IMHO) aside, they are really delving into quality technology based instructional activities that are way better than listening to some guy drone on in a grainy video lecture full of mistakes.  And they are doing it without financial support from the likes of Bill Gates and Bank of America (I'll let you read between the lines there).

Chutes and Ladders

I played a game of Chutes and Ladders today with my 5 year old, Jake.  Before we played the game, we were talking about numbers.  I asked Jake if he knew how to count by 10's.  He obliged by counting 10, 20, ... , 80, 90, 20, 21...

It was funny how he seemed to have these numbers memorized, but wasn't sure what to do after 90.  So I remembered this number grid we had at home...

I had Jake count by tens and mark them on the number grid.  He was pleased to see they all lined up and he grinned at remembering 100 came after 90.  Then I had him count by 5's and mark it on the grid.  Again, he noticed some patterns and overlap with the counting by 10's.  Then I let him pick any number in the top row and count by 10's from there, marking off the numbers as he went.  He caught on that he always made vertical columns on the grid when he counted by 10's.  I thought this was a nice start.

Then we played Chutes and Ladders...

The first thing that I noticed is that the organization of this grid of 100 was different.  Instead of the numbers looping to the next row, they wind their way back and forth on the page.  At first I was bummed and thought this new arrangement (as well as the busyness of the board) would just confuse Jake.  But what I noticed is that this game presented some very nice "what is bigger conversations".  For example, when one of went up a ladder or down a chute, we had nice discussion about which way will be going on the board now.  Jake landed on space 51 and climbed the ladder to space 67.  We had a nice conversation about which direction he should be facing in this new row.  He figured out that it amounted to which number was bigger, so he should point his token toward the 68.  It made for a nice greater than/less than conversation.

Thanks to Christopher Danielson for reminding me of this game, which we had tucked in the back of the game closet!

Math Thinking with Kids

One math teacher that I follow is Christopher Danielson.  Christopher makes regular posts on his blog about talking with his own kids at home about math.  I've found these posts to be very interesting, and have extended some of those same conversations with my kids.  I have three boys, ages seven, five, and two.  It is very interesting to talk through mathematical thinking with them and get a sense of how they see the world through numbers.

Here's a conversation my wife had with Aaron, the 7-year old, this morning while processing some summer practice subtraction he was working on.  He had been taught "borrowing" and "carrying" this year in school, and we wanted to see if 1)did he retain anything, and 2)what sense, if any, did he make from those procedural skills.

Mom:  Okay, so let's try out this one here: 32 minus 19.  How would you start this problem?

Aaron:  [after some pause for thinking, he starts counting on fingers] two, one, zero, negative one, negative two, negative three, negative four, negative five, negative six, negative seven.

He writes -7 under the problem.

Mom:  Okay, so negative seven is your final answer?

Aaron:  I don't know

Mom:  What do you do with the three and the one?

Aaron:  Three minus one is 2.

Mom:  Oh, is it just two?  What do the 3 and the 1 represent?

Aaron:  Oh, it's twenty

He writes 20 under the problem next to the -7.

Mom:  Nice, so now what?

Aaron: [pauses to think it through]  That makes it 13.

It was interesting to see his problem solving process here.  She went on to look at the "borrowing" method he learned last year, but really focused on the relationship between the tens and ones places and what it actually means to borrow (we busted out some Cheerios to help visualize it)

But it was very cool to see him work through this process.  Did he start subtracting 9 from 2 on the right hand side because that's where he's been told to start, or did he know he was comparing units?  When he had 20 and -7 on his paper, did it make sense for him to combine those values the way he did.  If the 7 had been a positive value, would he have added them?

When we teach kids procedural skills based on memorizing steps without attached meaning and understanding of numerical relationships, there's definitely a disconnect there.  It was interesting to see his wheels turning after 2 months of summer vacation.

3 Act Lessons

I'm in the works of developing a new lesson idea dealing with volume.  I've been participating in a website called 101qs (http://www.101qs.com/) where users post pictures or video of a perplexing nature in hopes of drawing out questions from viewers.  These questions provide the starting point for building a lesson around that picture or video.  This lesson building follows a 3-Act lesson style created by Dan Meyer.

Act 1 - the perplexing video or picture.  Ask students what questions they have about what they've seen. Have them make estimates about what they think the answer is.  Have them brainstorm a list of necessary additional information they think they will need to get to the bottom of things.

Act 2 - provide more information.  This could be a collection of data, additional pictures showing measurements that were taken, another video where students see that data collection and make their own notes, or a short lesson that provides the mathematics that are needed to solve the problem.  During Act 2, students work together to process the information they have collected to try to answer the question.

Act 3 - provide the answer.  This isn't the teacher saying "The correct answer is...."  Rather, you show them the answer.  The complete video that captures the end result.  A picture showing the final sum.  A link to a website that contains the answer.   Act 3 can be used to reflect on the process.  How close were our guesses, how accurate was our method of solving the problem, what follow up questions can now be answered or explored?

I recently went on a family trip to Wisconsin and we went canoeing/kayaking up a small river.  A few minutes into our paddling, we came across this lock and dam:

As we waited for the water level to drop so we could paddle in, I decided to take some video.  Here's the first take...

What questions do you have?

First Blog, First Post

Alright, here we go!  This is my first ever blog post.   Full disclosure, I'm writing this now because I have to.  I'm taking a class on blogs, wikis, and podcasts in the classroom.  This is one of my first assignments.

But honestly, I think creating my own math blog is something that I should do.  I have been following a number of great math teachers via blogs and Twitter for a few years now and I have gotten so much out of it. It's probably time that I contributed something to this "mathtwitterblogosphere" as well.  Problem is, I haven't felt like I have much to say.  At least nothing that will have a major impact on the global community of math teachers.  I'm more of a contemplative listener.  Sure, I've commented a couple of times on other people's blogs but there is a certain comfortable anonymity to that.  I haven't put myself out there yet with original thoughts of my own.  It's intimidating, especially considering the great ideas that are already out there.

So this is my first post.  I want to stick with it.  There won't be anyone reading this at first other than my course moderator and myself.  But I intend to keep at it after this course is over.  This will be my spot to reflect on what I'm doing, and if I have something great that is worth keeping and possibly sharing, that'll go here too.

So long for now...