Monday, September 30, 2013

K'Nex Linear Programming

I love linear programming.  It's actually applicable to the real world (huzzah!) and it's multifaceted enough to keep my interest.  I get bored of math problems that I can solve mentally without much thought.  Linear programming presents something with enough layers that I have to get out pencil and paper (or Desmos!) to do some of the calculations. 

The problem with linear programming is that my students don't appreciate the layers of the problem.  They view linear programming as too many steps and too much work.  They may not fully appreciate how useful it can be, even though they can completely understand that a business would like to maximize profit or minimize cost.

I tried to address this problem today with a small K'Nex experiment.  Essentially, I wanted to give the kids the resources of a linear programming situation and ask them to solve it without the system of inequalities.  The handouts I used are embedded.  They're sized for composition books. 

First, I had students attach the scenario to their notebooks and create a table.  You can see this in the photo below. 

I gave each group a bag with the amount of K'Nex specified in the question and asked them to build combinations of "bots" and "quarks" based on the restrictions in the scenario.  Most groups latched on to the idea of needing to use as many pieces as possible, so they didn't record many pairs that had lots of left over pieces.  After about 15 minutes of building and recording, I had groups report out.  There were a couple of cases where students didn't agree about the maximum profit, but in every case they were able to find their error.  Either they used more pieces than they were given because they didn't physically build the models, or they made a simple calculation error. 

A hint if you try this: organize your bags so that a group has only one length rod and one shape connector.  I had some students who were trying to make "different bots" by connecting the pieces in different ways.  I had to review the definition of a bot with them and explain that the construction didn't make a difference as long as the materials were consistent.  I think this issue would be muddled by offering different shapes and sizes of K'Nex to one group.  It's also helpful when a piece ends up on the floor as you can more easily glance around and see which group it might have come from. 

I heard many students say they enjoyed the activity and from listening to discussions around the room, I feel like I introduced this in a way that was concrete enough that most of my students grasped the concept of linear programming before we got to the algorithm.  Yes!

Tomorrow we'll use the other pages from the embedded file to take some notes of all the steps of a different example and we'll use that process to verify our solution from today's exploration.   In my experience, the hardest part for students is choosing the variables and writing the inequalities for the constraints. 

What's your favorite toy to bring into the classroom?

Mathematically yours,
Miss B

Sunday, September 29, 2013

My best tip on Interactive Notebooks thus far!

I am loving using Interactive Notebooks by a large.  The one thing that does bother me is that my kids are still not all that fast at getting things assembled.

Tonight, I came up with a simple solution to keep things moving.  Legal paper!  I've heard that other teachers like to use half sheets, but I don't think a 5.5" by 8.5" piece of paper is quite enough space for most things I want to include.  I realized, though, that legal paper would cut in half to 7" by 8.5" which just leaves a small margin of extra space when the sheet is glued into the ISN.  All I'll need to do is to format the handouts two to a page and chop them in half.  Huzzah!  Now, to find out if the school has legal paper hiding anywhere...

To give you size comparison with a regular composition book.  I didn't attach it because I don't have my model notebook at home, but it leaves about 3/8" on each side and the bottom with a bit extra up top so there's room for a title. 

Here's a KWL chart that you can print on legal paper.  Font is Arial Black which should be standard for almost everyone.  I used KG Shadow of the Night on mine, but not everyone has it installed.  :)

What's your best time-saving ISN tip?

Mathematically yours,
Miss B

Saturday, September 28, 2013

Using Visual Patterns to Introduce Linear Functions

A few months ago, I found and I knew it would be useful as I started linear functions with my 8th grade students. 

Here's how I've used it.

First, we learned what functions are and practiced completing function tables given a function.  My students are weak in computational skills, so that was important to tackle before we got too deep into  linear functions.  Then, we used the steepness of stairs to explore the concept of slope.  Once we'd learned to find slope from a graph, table, two ordered pairs, and word problems, I was ready to introduce linear functions.

I had students start with pattern #2.  They built the pattern with unifix cubes and completed a recording sheet.  The recording sheet asked them to find the number of cubes used in the first six terms of the pattern and then it jumped to the 15th term.

As I circulated around the room, I saw some students were building to get to the 15th term while others noticed the numerical pattern and chose to complete a table.  I also saw several students who were off by one or two blocks on their answer.  One student had written 30 (the correct answer is 29).  We had a rich discussion which I'll try to capture here.
T: So why is the 15th term made with 30 blocks?
S: Because it's 15 and 15. (She gestures horizontally and vertically.)
T: Can you build it and show me?
S: Like this.  Student builds an L shape with approximately 12 blocks on the horizontal and 7 on the vertical.
T: Would you explain how you made that?
S: I just added some blocks on.
T: Why did you add that amount of blocks?
S: I don't know; I just added some.
T: OK.  Did you notice a pattern happening when you built the first four objects?
S: You put one on each end.
T: Could you try that here to get to the 15th term?
S: Yeah.  Student builds correct 15th term as we count out loud which term number she is on.
T: So, how many blocks did you use?
S: 30.  (Student answers immediately without counting the blocks used.)
T: Could you count them?
S: But there's 30.
T: Please count them so we can be sure.
S: OK.  1, 2, 3, ..., 29.  That's not right.  1, 2, 3, ..., 29.  Hmm.  Student picks up another block and adds it to her construction.  30!  There's supposed to be 15 and 15. 
T: Why are you changing what you built?
S: Because it's wrong.
T:  Before we decide that it's wrong, let's look at a smaller object.  Look at the third term.  How many are there?
S: 6. (immediately)  Oh, no.  There are 5.
At this point, the student sitting across from her can't take it any longer.  He interjects his observations.  
S2: Look, if you take the L apart, there are two equal lengths with the one block in the corner as a connector.  Student 2 models what he's describing with the blocks by taking the L apart and showing Student 1 that the sides are equivalent if you remove the block in the corner.
S1: OK, but why isn't it 30?
S2: Because you have to have that corner piece, too.  
T: Have you noticed any mathematical pattern in the first few terms?
S1: It's adding 2 each time.
T: And do you notice anything about the kinds of numbers that are in the table?
S1: They're all odd.  So 30 couldn't be on the list.

Our conversation actually went on a bit more, but I wanted to capture this part because I thought it was really good.  I was a bit shocked at how difficult this process was for Student 1.  I was also pleased by Student 2's observations as I suspect he'll do well as we make the connection to slope and y-intercept next week.  I think next week I'm going to ask Student 1 to use a new color for each term so she can see clearly what she has added in each step.

As we wrapped up this pattern, students graphed their data points.  I asked them what they noticed.
  • It's a line.
  • It's positive.
  • It's a function because it passes the Vertical Line Test. 
No one said, "the slope is 2" but they'll certainly catch on to that next week as we examine a few more patterns and write down our observations. 

Edited 7/23/15 to add my recording sheet:

And a second version I used with Algebra I:


Thanks to Fawn for offering such a helpful resource! 

Wednesday, September 18, 2013

Let's see if I can make this coherent...

We're now four weeks into school and I've been using interactive notebooks with my students for the last three weeks.  They are new to me and to my students, so I expected some growing pains. 

Today, two of my students told me that they prefer the way they took notes last year (copious notes on looseleaf).  I'd be lying if I told you I wasn't a little crushed.  I left out my disappointment when I talked to them about it.  I asked why they preferred the notes last year and it seemed to boil down to the fact that they like direct instruction with lots of examples.  I thanked them for their input, said I would take it into consideration, and told them I appreciated them talking to me about it because I know it takes a lot of courage for middle school students to have that kind of conversation with a teacher. 

I have to say that I don't disagree with the girls that a few more examples over the most recent topic (solving systems of equations) would have been helpful for some students.  The thing I'm now trying to figure out is how I make that work with the interactive notebook.  I don't want to have students take lengthy notes only to later condense them in the notebook because that would take so much time.  I also don't want to do a few examples on a foldable that is saved and a few others on a sheet of looseleaf that won't ever be seen again. 

My other issue with this comment is that my school is trying to ever so gently move away from direct instruction.  While it has its place, we know that there are lots of other ways that students learn and express their knowledge that are much more dynamic and engaging. 

If you use Interactive Notebooks, how do you find the right balance between overly condensed notes and lengthy notes?

Mathematically yours,
Miss B

Monday, September 16, 2013

Notebook Check

It's time to start checking the contents of students' Interactive Notebooks.  My plan is to check them roughly every three weeks beginning this week.  I have just shy of 80 students, so I'm planning to check 20 notebooks Monday through Thursday during a normal notebook check week and to get any stragglers on Friday.

I shared a rubric during the summer that I'm using with my classes.  Today, I guided my students through their notebooks and pointed out the portions that needed to be complete prior to my checking the notebooks.  Several students realized they had a few portions to complete or improve before their first grade. Tomorrow, we're going to all mock grade our notebooks before I collect any.  I want the students to grade their notebook and then see if my grade matches theirs. 

One expectation that I have for my students is that if they do not earn full credit for completeness on a notebook check, they must catch up prior to the next notebook check.  In other words, if they didn't have pages complete for notebook check #1, those pages will roll over into notebook check #2.  Double jeopardy?  Perhaps.  However, I feel like it's the best way to truly hold students accountable for staying on top of everything. 

To help with this notebook check process, I posted this sign in the classroom.  The reverse side says "next week" so I can post it prior to the notebook check also.  The main thing here is to include the page numbers so students have direction as to what they need to look over prior to notebook checks. 
This is my #Made4Math Monday entry.  I think it's been three or four weeks since I've posted one, so I'm happy to have something to share, however small.  Want the file?  Here it is ready to print. 

Mathematically yours,
Miss B

Tuesday, September 10, 2013

Day 11- Now for something totally unexpected

When I created my list of left-side assignments this summer, I assumed that some would be more popular than others because kids would assume that they were "easy."  I did not expect that the first time I assigned a choice of left side assignment that I would get items of as high a quality as I did. 

Here's a song about Standard Form, to the tune of Justin's Bieber's "Baby." 
I was also impressed by a couple of comic strips, a set of flashcards, and by the student who typed his page-long assignment with one hand since no one was home to help him and he is in a giant splint following a sports injury.