Wednesday, March 26, 2014

4!

Me: I need two volunteers. You have no idea what you're doing. Thanks Brianna and Jesus. Go stand in front of the whiteboard on the side of the room. You are the two contestants in today's Spelling Bee.
This is how I opened today's lesson. Wait. A Spelling Bee in math class? I address the audience:
Me: I need your help. I am going to ask you a question. The answer is a number. I am not interested in any categories like gender, height, age, birthday, first name, last name, etc. For my Spelling Bee, I need you to take my contestants and order them for me. What's the maximum amount of ways I could order these two contestants?
Students have time to think and some quickly raise their hand to say, "Two."
Me: Show me. Tell us what they are.
Student: Right now Brianna is first. Jesus is second. We could switch them and Jesus goes first.
Me: [looking at Brianna and Jesus] Do what she said.
Brianna and Jesus switch order.
Me: Have I maxed out all the possible combinations for ordering Brianna and Jesus?
Class: Yes!
For a little comic relief, I toss Jesus an easy word to spell.
Me: Jesus, spell "cat".
Jesus: C-A-T
Me: Wait. What?
I learned today that most kids don't know how a spelling bee works, so I call on a few kids to explain the three steps:
1. Say the word.
2. Spell the word.
3. Repeat the word.
Me: Jesus, let's try this again. Spell "cat".
Jesus: Cat. C-A-T. Cat.
Me: Bri, spell "discombobulate".
Brianna: Ughhhhhhh. What?!
Me: Okay, can I get a third contestant for our spelling bee? Jesus, since you're the winner, please pick someone.
Standing in front of the audience, I now have Jesus, Brianna, and Garry.
Me: Okay, let's say their current order is one possible combination. Let's keep Jesus first. Can you get any other combinations with Jesus being first?
Student: Yea, switch Bri and Garry.
I look at Bri and Garry.
Me: Do it! Okay we now have two possible combinations. Have we maxed out the possible combinations with Jesus being first or can we get more?
Class: We're maxed out.
Me: Okay, someone give me a new combination.
Student: Put Brianna first this time. Then Jesus. Then Garry.
Me: Okay, we now have three combinations. Can we get more where Brianna is first?
I repeat this process until the class has agreed we maxed out our combinations with six total. Great. I toss this information in a table like this to keep track of it.
Me: So what if I add a fourth contestant to the spelling bee?
Sarah: No!
Me: Really Sarah? What? Are we going to have more or less combinations?
Sarah: More.
Me: Gimme some guesses everyone. Toss something out there for fun. How many combinations could we get with four people in the spelling contest?
Students tell me 8, 10, 9, 12, 16, 13 and I write all of them up on the board. I ask for some quick reasoning behind the guesses.
Me: Ok, thanks. You all can't be right. Instead of moving people around, let's do this instead.
I gave each group a sandwich bag with four different colored snap cubes: red, green, blue, yellow. Students were to work in their groups to figure out all the possible combinations of four colors. They were to write it down in their notes for the day. I circulated the room, noticing student work.

For groups that think they're done, but wrong (like only 12 combinations):
I zone in on one combination and keep their two colors fixed, "Have you maxed out all the combinations with these two at the front?" Usually this is the only nudge they need to get closer to the correct number of combinations.

For groups that are on track:
I make it obvious I note their work, or ask for a quick explanation, or I quickly move to another group.

Groups that finish and have the correct answer:
I have them explain their work, organization, process, and reasoning. I ask if they feel confident and usually they do. I'm not going to string them along. I respond, "That makes sense to me." followed by:
Me: So what if I gave you a fifth color?
Student: [typical response] Ughhh.
Me: Oh, what's wrong?
Student: That's a lot of work.
Me: I know, right? I'm right there with ya. I wouldn't want to write out all those possible combinations either. So, your job is to try and figure out a shortcut. In other words, if I just gave you four colors right now, how could we quickly get 24 combinations without writing them all out. If I'm now giving you five colors, what would be a quick way to figure out all the possible combinations?
Once I see that most groups have reached the magic number (24), I show them this and have them count.
Me: One clap on three for the closest guess.
1-2-3 CLAP!

Many kids see that 4 groups of six combinations yields 24 combinations. I toss 24 into our table and ask the whole class about finding the possible combinations for five colors. Typically, the students want to avoid this nonsense and express some noise of rebellion.
Me: What's wrong? You guys don't want to write out all the combinations? Well, let's try and find a shortcut. Do we see anything from our table that might help us?
To my pleasant surprise, at least one kid in each of the three participating classes found the following relationship:
Abraham, Brianna, and Daisy: You take the previous "Combos" result and multiply it by the diagonal "Colors" amount to get the new amount of "Combos."
Me: Let's see if that works.
It does. Great!
Me: Okay hot shots! This is a great shortcut. What if our principal walked in and gave us 13 colors. How would I quickly figure out the total number of combinations since I don't have the number of combinations from 12 colors?
Here's where I introduced the use of factorials. Yes, I could have spent time getting the kids to look for this pattern, but I simply didn't have or make the time. I felt it was a good place to show them that putting the factorial symbol after a number means to multiply it by all of the natural numbers less than the given number.

4! = 4 x 3 x 2 x 1 = 24
Me: So if our principal walked in and said, "Find all the combinations of 13 colors." we'd go thirteen...
Class: ...times twelve, times eleven, times ten, times nine...
In reflection, this lesson created more successes for my students than I anticipated. Some include:

• Discovering patterns and relationships within a table,
• Creating a need for the factorial of a number,
• Adding another vocabulary term to our tool belt, and
• Finding combinations more efficiently.

This lesson started with a low-entry of two students and two combinations.
We built in the next part by finding six combinations for 3 students.
We built in a guess for the combinations of four students so they can invest in the question and look for patterns.
We manipulated four colors, organized our combinations, made conjectures, and arrived at a reasonable answer that maxed out the combinations.
We pushed those students who finished early to discover a shortcut on their own.
We created a need for avoiding excessive work with larger numbers and a need for some type of formula (factorials) that will get us the same result.

I came into this lesson with a rusty understanding of factorials, probability, and combinations. Anyone who is against Common Core State Standards, think again! It's making math teachers know their content better, so they can better serve their students. It's opening the door for students to reason their way in math class. I'm not blogging to get into the importance of CCSS right now. However, I'm convinced this was way better than me standing in front of the students telling them to put an exclamation point after 4 (like this 4!) and to just multiply 4 by 3 by 2 by 1 to get all the possible combinations of four somethings. Instead, the students discovered the relationship (pattern) within the table and felt confident in discovering the total combinations of five colors without drawing them all out.

Factorial,
848!

Monday, March 24, 2014

No matter who you are, who you teach, what you teach, and what type of school demographic you teach, teachers always have to account for student behavior and classroom management. Likewise, you might be the most engaging teacher, have the most awesome lessons, and/or have a lot of students who adore your every sneeze, but we can all benefit from keeping track of student progress. Enter my crush on Google forms/docs/drive this semester for two reasons.

After-School Help
Nothing complicated. When students show up after school for math support (voluntarily or involuntarily), I have a quick way to keep track of who showed up and how long they stayed. The "miscellaneous notes" section is helpful for tracking students' skills or questions they might have, etc.
This can also be helpful when working with parents if you have an action plan for their child to receive additional support outside of class time. The last thing I want to do is create more work with these forms. When students show up, I have them write their name on the board and the time they depart so I can quickly enter their work time. This is a short and sweet form. Let's move on.

Behavior Log
The purpose of the form is to log any interaction I have with a student as a result of being off-task, misbehaving, or anything else that disrupts the learning process. The purpose of this Google form is not to curb bad behavior. However, I will say it can be effective to fill out the form together with the student. Tread lightly: don't make a show of it in front of their classmates or project it up on the screen for all to see.

Behaviors:
This is not an exhaustive list of classroom (mis)behaviors. However, think how easy and efficient it is to check common behaviors. When filling this out with the student, it helps to have them identify what behavior disrupted the learning process. The "other" option takes care of anything you can't foresee your students doing. Always nice to have.
Action Taken:
My school expects teachers to handle as much classroom management issues as possible by having us layout a progressive discipline procedure with our students. As you can see, my list under "Action(s) Taken" seems pretty progressive, or at least I think so. For me, the most meaningful and effective action is the "Student-teacher conference." Whatever your fancy is, create a list of actions you usually find yourself doing and make them checkboxes. Don't forget the "other" section.

If this happens again...
I have a really porous memory so this section is a lifesaver. You're telling the future you what to do if a student repeats their behavior. I can't tell you how many times I just open the Google responses for my log, press Command-F (for find), type the student's name, and BAM! I have what they previously did and what we agreed on as the next step in progressive discipline.

I sometimes use this to make a note about the student responding well to a warning, the details of a student-teacher conference, or the actual incident itself. It's there for what you need it for.

Final thought:
Create a shortcut in your browser for these Google forms. If you're out in the wild with an iPad, create a shortcut there too.

Don't get me wrong people, I'm not bragging about student discipline with this post. I believe that most student misbehavior can be prevented by providing students meaningful/engaging learning experiences, classroom boundaries, and routines. Mix this with a lot of preventative-maintenance teacher moves and students typically stay on task and out of trouble.  But we can do more than that.

PBIS
Our school has also required every teacher to include a Positive Behavior Incentive System (PBIS) in their classroom. Students earn some type of token for positive contributions to the learning environment and can cash them in for prizes that range from candy, to sitting in the computer chair, to a bag of chips, to an Expo marker, to picking something from the mystery box.

Students used to earn stickers in my class for positive behavior, where they could cash in the stickers for prizes. It was a hassle for all of us. Recently, a colleague went to a conference and shared a PBIS idea I've found to be pretty effective. I hand out small "Thank You" notes printed on scratch paper. Students save them and can cash them in. So far, so good.

My goal of this post was to encourage you to look into Google forms for efficiently keeping track of student interactions. If you have others, please share.

Crush,
1014

Sunday, March 16, 2014

Carnival of Probability

This past week, my awesome partner, Hannah, and I hosted a Carnival of Probability for our 7th graders in our school's multi-purpose room. Let's get to it:
Station 1: Spinner 1
Pick a side. Will the spinner (arrow) land on the 5 or the 15? If you choose wisely, you win that many tickets.

Station 2: Spinner 2
No need to pick a section. You get three spins. Land on the 1, you get one ticket. Land on the 100, you win 100 tickets. BAM!

Station 3: Rolling a die
Pick a game. Grab the six-sided die and roll.
• Roll an even number, win 8 tickets.
• Roll a multiple of 3, win 15 tickets.
• Roll a one, win 30 tickets.

Station 4: Rolling two dice
Pick a game. Grab two dice and roll.
• Roll two sixes, win 45 tickets.
• Roll an even number and a five, win 45 tickets.
• Roll two multiples of three, win 32 tickets.
• *Roll the same number on our twelve-sided dice (pictured below), win 70 tickets.
 See the blue 12-sided die? There's a 12-sided die inside. Cool, right?!
Station 5: Bag of letters
Pick a game. Reach inside the bag of 26 chips chips (pictured above) and pick one chip.
• Pick a vowel, win 25 tickets.
• Pick a consonant, win 21 tickets.
• Pick the first initial of your first name, win 40 tickets.

Station 6: Ball toss
Take a ball. Toss it in the direction of the cups. If it goes in any white cup, win 3 tickets. If it lands in the red cup, win 35 tickets. Just like that!

Station 7: Deck of cards
Decide on a game. Pick a card or two. Win tickets!
• Pick a red or black card, win 1 ticket.
• Pick a red card and a queen, win 40 tickets.
• Pick a Jack, Queen, or King and win 22 tickets.
Station 8: Coin toss
• Flip it once and land on heads, you win 8 tickets.
• Flip it twice and land on heads both time, you win 20 tickets.
You might ask, "Did you really pass out all those tickets?"
The simple answer is, "No!" You don't think we're that crazy, do you? We assigned one to two students as captains for each game. The game captains gave each contestant one ticket with the number won written on it.

The math: we spent a few days leading up to the carnival talking about probability and identifying "and" statements along with "or" statements. Station 4 was all "and" probabilities which made your chances of winning more challenging. The 12-sided dice game had a less than one percent chance of winning. Station 7 and 8 had a couple of "or" probabilities. Our goal here was for students to have a concrete introduction and application of probability. To enter the carnival, students had to represent the probability of each game as a fraction, decimal, and percent.

The students had a blast. It was fascinating to hear from them about the carnival the following day. They explained which games were the "easiest" and "hardest." The games (as you can see from the pictures) were nothing fancy, but they did the job. Students will get to cash in their tickets for prizes such as candies, pencils, pens, and Expo Dry Erase Markers. What would your carnival look like? Please share.  Head over here for the paperwork/handouts we used.

Carni,
633