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."

Folders,
918

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,
336

NEW JOB!!! and some fraction ideas

I recently accepted a new teaching position with a middle school where I'll be teaching 6th and 7th grade math. I was fortunate to be at my last school for about 10 years exploring 7th and 8th grade math: Pre-Algebra, Algebra 1, Algebra 1A, Algebra Honors, and Geometry. I'm extremely grateful for the opportunities, experiences, friendships, and professional growth opportunities the school afforded me. As I advance in my teaching career, I'm very excited about my new position, new school, new students, and new everything. There are many differences between my previous school and my future school... and I welcome them wholeheartedly.

As my future school transitions to Common Core, I'm giddy at the thought of exploring so many wonderful concepts in 6th and 7th grade math. However, I will be working with students that have typically struggled when it comes to understanding math. Therefore, I had a few ideas about fractions I thought I'd like to explore with you.

I'll include all the visuals here, but feel free to go to my "fractions test page" at Estimation 180 to get the full experience. Please offer me some feedback. I'd like to pursue these "fraction" ideas with other items; some easier, some more difficult. Is this something you could use? Is this something worth pursuing?

Question: Where would the cylinder be one-third full?
(Image 1)

We're estimating here. I did not provide any choices because I want students to formulate ideas on their own. Look at their screen and move their finger up and down the screen to find one-third. Come up to the board at the front of the classroom and put a post-it note on the board.

Offer some choices: When ready, click on the image for choices.

Notice I said, "when ready"? Did you have students discuss? point with their fingers? place a few sticky notes on the screen at the front of the room? or something else to get students invested? Because of the restrictions at Estimation 180, this image will currently serve as the next viable step. Now students have a choice. I'm not the biggest fan of this, but it's something. Were there students who were way off because their sticky or initial guess didn't even fall within the given range?

Make a choice and demand reasoning: Why did a student choose "C" instead of "D"? Have students try and convince each other. Argue! Egg them on a little bit. Have students choose a line in which they think the cylinder will reach one-third its capacity.

Do some math? I provide you with the capacity of the glass: 1,170 milliliters. Find one-third of that. Encourage different strategies in your class. Doing the math won't tell students if the answer is choice A, B, C, D, or E, but it might help with later parts of this activity.

Reveal the answer: a really short video.

I have additional video for two-thirds, fourths, and a full cylinder (when using thirds or fourths). I haven't inserted the choices, added a counter, or other after effects. Would this be something you'd be interested in? Please let me know.

Two things:

1. I also set this up as Red Dot (Active Prompt) activity and it'd be fun to see how students would approach this activity without multiple choice. Then, show the class their results before watching the answer (video).
2. I'd love to see Dave Major make a slider so students could slide a bar up and down the cylinder. Using a computer or tablet, students could place the bar where they want and without a given range of choices. Then we could see who was actually correct.

What feedback do you have for me? Again, is this something you could use? Should I prepare more at Estimation 180? Would you like to see the remaining fractions and other ideas?

NEW,
723