Manilla folders [Number Sense]

I must share this short story with you about the number sense I've been witnessing for the past few weeks with many of my students. Be warned: not for the faint of heart.

Today, a few 7th grade boys came in after school today and wanted to help with some things around my room: tidy up desks, organize whiteboards, etc. There was a box of 100 manilla folders that I had just brought back from the office. I needed 80 of the folders and would later return the remaining folders to the office.

Here's my exchange with the student we'll refer to as Albert:
Me: I need 80 folders from that box. Albert, think of a quick way you could get those 80.
***Let's pause. How would you (the reader) quickly get 80 folders from this box?
Albert: I could count in 5's.
Me: Okay. Any quicker than that?
Albert: By 10's.
I get what Albert is trying to do. He doesn't want to count to eighty by ones. To that point, I would say his initial two responses made sense and are practical, in the mind of a 7th grader. I thought maybe I'm asking the question incorrectly, so I try it again.
Me: Right. That would be a good way to organize the folders to keep track of them as you count. Albert, I'd like you to think of a way to quickly get those folders out of the box. 
I can see the look on his face is one of confusion. Not that look like he's trying to figure it out, but that look like he has no idea what I'm asking. So after a minute, I mistakenly ask another question (in hindsight, I wish I would've stopped the conversation and let him do his thing).
Me: How many folders are in the box?
Albert: 100.
Me: How many do I need?
Albert: 80.
Me: Is there a way to get me the 80 without counting all 80?
Albert has no idea. I like this question because now it's specific. His challenge is to get me 80 folders without actually counting all 80. After a minute. He needs some prompting.
Me: If the box has 100 and I need 80, how many will be left?
Albert: 30.
Me: So 80 plus 30 is 100?
Albert: No wait, 20. 
Me: Okay, so I will send 20 back to the office. How can I get 80 folders out of the box without counting all 80?
Albert has no idea. This exact exchange continued for another round. I'd like to say that Albert eventually came to an efficient way on his own, but he didn't. I tossed 100 up on the whiteboard. We subtracted 80 and wrote 20. I thought after Albert saw the 20 on the board, he might realize to count 20 folders from the box, take out the remaining folders and switch them with the 20.

This is quite common. The number sense (or lack thereof) my students have (or don't yet) is quite fascinating. I have a lot of work ahead of me this year. One thing is for sure. I can get better at asking shorter questions. I can get better at looking for these learning opportunities. I can get better at not looking for one answer when asking questions of students. As Max Ray would say, "2 > 4."


Your BF!

The first full week of school is done. I can breathe... a little.

I'd like to share a lesson we (my awesome 7th grade teammate and I) came up with this week when reviewing multiplication and division of fractions with 7th grade students. It actually went quite well and was fun. Simply titled, "Your BF!"

If we said "fractions" to students, a majority of them would instantly shut down. Instead, I titled their notes at the top with "Your BF!" and it looked like this:

Of course, some kids, only reading the title, responded with:
"Your best friend? What!?"

Then, some kids continued to read and saw it. That word. That terrible, miserable word. That word that has a paralyzing affect on students. Almost like you just told them their dog was run over by a car, or that Christmas was cancelled, or [insert some terrible event you dread]. That word is FRACTION.
I quickly responded to their whining, groans and moans by saying, "Let's make this a little friendly competition. Let's see who can draw the best half. Use the rectangle I gave you to draw your best half."

All of a sudden, it's a game. There's a challenge.  No rulers or measuring devices allowed. They must freehand the fraction. As they're working on their BF (Best Fraction), I announce, "I'm looking for two volunteers who think they've drawn the best half."

Immediately hands shoot up. Kids start to yell at me, "Pick me Mr. Stadel!"

Everyone feels so confident about drawing a half. I yell over them, "I'll put it up on the document camera and you all can be the judges today."

"Oh! Pick mine."
"Me! Me! Mr. Stadel, pick me!"

I pick two students and they walk their papers up to the camera. The class is critiquing each classmate's halves. Some students, from their desks, quickly blurting their half is better. I then display mine (having used a ruler) just so they could see an ideal half. This is where I wish I took pictures of student work. We put it to a vote. I noticed some kids just voted because their friend is up there or they might not like one kid, or whatever. Therefore, in hindsight, I should have specified that they vote for the fraction who's line looks closest to the middle and splits the rectangle into two equal parts. Even more hindsight, I should have asked students what they think would constitute a half. Next time!

Repeat the competition with a best third and a best fourth:

We compared three student papers for the third and four student papers for the fourth. After our competition, we transitioned into the visual representation of multiplying a half by a fourth.

At first, this visual seemed a little confusing to some students, but we walked through it together. When circulating the room to check for understanding, I could see better understanding on the second example in which they completed on their own.

After all of this, here are a few favorite moments:

  1. When moving onto the best thirds, I heard a student say, "I don't get this." They must have looked at their neighbor's paper. Within seconds they responded with, "Oh, I get it."
  2. When showing students my half, I accidentally showed them the rest of my paper with the ideal thirds and fourths. They responded, "Ohhhhhh!" Like they just saw the answer key or it ruined the fun for them to see the answers. That was funny. 
  3. One student's paper was not picked, and he turns to a friend and I overhear him say, "He should have picked my thirds. Look at that sexy fraction." I stopped class and he thought he was in trouble. Nope. We had a good laugh as I shared with the class that I've never, in my ten years of teaching, heard a fraction described as sexy. 
  4. When doing the visual representation of multiplying, I said, "Add this to your notes." A student replies, "Oh, we're taking notes?" He was participating and didn't even know he was learning.

Here's the handout, which also includes an example for dividing fractions.

This lesson idea spawned from my Best Halves idea.

Your BF,