Sunday, October 6, 2013

MTBoS Mission #1: A favorite problem

Hi to anyone who's visiting thanks to the Explore MTBoS blogging event.  If you haven't heard of MTBoS, it's short for Math Twitter Blogosphere which is essentially a bunch of math teachers who like to connect online and share their best resources and ideas to advance the profession.  Or something like that.  It's loosely organized and self governed, so there are no official rules!

Anyway, welcome and thanks for reading.  I've been blogging about my professional life for a little more than a year.  You can find me here at and also at my 180 blog where I try my best to put up a short post from my classroom each day.  I'm also on twitter, handle @iisanumber. 

Sam asked us to respond to this prompt as our first of eight weekly challenges:

  • What is one of your favorite open-ended/rich problems?  How do you use it in your classroom? (If you have a problem you have been wanting to try, but haven’t had the courage or opportunity to try it out yet, write about how you would or will use the problem in your classroom.)

I chose a semi-open ended problem, but one I like very much.  I think I first wrote it about three years ago when it was a completely bland word problem.  For your comparison, I left that page as the third page of the document.  It harkens back to the days of ECRs (Extended Constructed Responses) in MD curriculum.  It's interesting to look back on old things and revise them!

In the problem, students are first presented with two competing movie rental plans and asked to compare them.  An improvement I made to the original problem was that I presented the data in different forms so students have to work from a word problem and a table simultaneously.  After students compare the two plans, a third plan is introduced in the form of a graph and the students are asked to make further comparisons.  Finally, they are asked to create their own plan and advertise it to the appropriate customers.

I used this task a few weeks ago after my students had worked on a quiz about systems of equations.  I liked that different students chose different representations with which to compare the plans.  It was a nice way to get a secondary assessment of how my students did on the systems unit.  I noticed that their most common issue was figuring out the equation based on the table.  Many of them erroneously chose the cost of one DVD per month as the y-intercept instead of recognizing that the monthly fee is based on renting 0 DVDs. When I noticed the error in their equation, I would ask them to verify their equation using a number of months on the table and each time they noticed their total was too high and they were able to reduce the monthly fee to the correct amount. 

The main reason I chose to write my post about this problem was a comment from one of my students.  T said to me after he turned in his work, "You know, Miss B, at first I didn't really get the problem.  But I sat and thought about it for a while and tried some things and then it just made sense and I could do it.  It challenging but not too hard."   I like to strike that kind of balance whenever I can, and I thought I should share this problem because not only did I feel like I was expecting a certain level of rigorous thinking from my kids with the task, but they articulated the same thing to me.  

If the file would be helpful to you, you are welcome to download it from below. Let me know if you have any feedback that could make it better.  Thanks!

Sam's second question was all about what makes your classroom unique.  I won't go in depth with my answer since we were asked to choose just one prompt, but I think this post sums up my answer AND backs up my colleagues' assertions that I am a math nerd.   :)

Mathematically yours,
Miss B

Friday, October 4, 2013

Distributive Property and Combining LIke Terms

Thanks to Julie and Nora, I was inspired to introduce combining like terms and the distributive property to my students using some manipulatives.

I started with a grocery bag (one of the ubiquitous freebies emblazoned with a company logo), a dozen ziplocs, some buttons, some paper clips, and some cap erasers.

Like Nora suggested, I set up some ziploc bags with a few items and I made several identical ziplocs.  To make it easy for my students to differentiate between the bags from a distance, I placed a piece of colored card stock in each bag.  The set up is illustrated on the ISN page below. 

Students volunteered to come up one at a time to place items in the grocery bag or remove them from the bag.  They could only touch one kind of item per turn (so Ziplocs with yellow paper, loose buttons, Ziplocs with red paper, etc).  We started with combining like terms using just the loose items and tried three practice rounds.  Each time, I asked the students how many of each item were in the bag before we showed how to write the answer algebraically.

Then I introduced the Ziplocs to the activity.  We discussed how parentheses group items in math just like the clear bag was grouping the items.   Some of my students erroneously thought that we could show multiple identical bags by using an exponent but with some follow up questions they decided that multiplication was what they really needed.  Again, as we wrote our expressions, I had students predict the simplified expression before we wrote the work algebraically.

The real magic happened when they had a homework problem like 9[5 + 2(x + 3)].  I asked, "if the parentheses are represented by the ziploc bag, what would the brackets represent?"  They told me the brackets were modeled by the grocery bag.  "How many grocery bags would we need?"  They decided they would need nine grocery bags.

I wish I had done this activity using pennies as well.  If I had, I would have made them each worth 1 instead of assigning them a variable.  One point of confusion I saw after this activity was what to do with an expression like 12 + 5x.  Many of my students tried to make that 17x and I think we would have done well to have constants in the original lesson.

Thanks to Julie and Nora for the inspiration.  I was so excited about this lesson that I immediately shared it with my colleagues.  This is the best part of the MTBoS- great lesson ideas at your fingertips, tried out by real teachers!  

Mathematically yours,
Miss B