# Making them think!

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I had a very cool lesson today with my sixth graders and and I just had to share. So often we say that our kids don't know how to think on a higher level. Let's share what we do to bring them to that level. Whatever subject you teach, post an idea.
Here's what I did today. It's not much, but the way the kids responded just got me really jazzed.
6th Grade Math:
As the kids came in to class, there were three warm-up questions on the board:
What is the probability of:
1) Tossing heads on a fair coin?
2) Tossing heads twice in a row on a fair coin?
3) Tossing heads three times in a row on a fair coin?
After going over it and discussing concepts like 1/2^3 as the probability for the third question, . I put on a scene from the movie "Rosencrantz and Guildenstern are Dead".
Essentially, the scene begins with Guildenstern finding a coin on the ground. He tosses it and it comes out heads. He does it multiple times and it comes out heads every time. After about 50 sequential heads, the scene continues with this interchange:
GUILDENSTERN: It must be indicative of something other than the redistribution of wealth. He flips a coin to Rosencrantz, who looks at it.
ROSENCRANTZ: Heads.
GUILDENSTERN: A weaker man might be moved to reexamine his faith. If for nothing at least in the law of probability. He flips another coin to Rosencrantz.
ROSENCRANTZ: Heads.
GUILDENSTERN: Consider. One. Probability is a factor which operates within natural forces. Two. Probability is not operating as a factor. Three. We are now held within un- sub- or super-natural forces. Discuss.
ROSENCRANTZ: What?
GUILDENSTERN: Look at it this way. If six monkeys … if six monkeys … the law of averages, if I’ve got this right, means that if six monkeys are thrown up in the air long enough, they would land on their tails he throws another coin to Rosencrantz about as often as they would land on their
ROSENCRANTZ (looking at the coin): Heads. Getting a bit of a bore, isn’t it.
GUILDENSTERN: A bore?
ROSENCRANTZ: Well …
GUILDENSTERN: What about the suspense?
ROSENCRANTZ: What suspense?
GUILDENSTERN: It must be the law of diminishing returns. I feel the spell about to be broken. He flips a coin high into the air, catches it, and looks at it. He shakes his head. Well, an even chance.
ROSENCRANTZ: Seventy-Eight in a row. A new record, I imagine.
The scene continues, but you get the gist.
After the end of the scene, I started the discussion.
"Based on today's warm-up, what would be the probability of tossing heads 78 times in a row?"
They came up with 1/(2^78). (Of course, someone asked if that was the same as 78^2, which was a great 2-minute tangent.) We started to show how ridiculously large 2^78 actually was (we made it as far as 2^12 before deciding that it was just a big number) and that 1 over that big number was really small number.
I then asked, "If I've tossed heads 77 times, what's the probability of tossing heads on the next toss?" Some of them felt that it was still 1/(2^78) while some saw that it was an independent event and that the probability would still be 1/2.
Now, I do this activity every year and this was actually the first time that somebody didn't say, "100%" and back it up with an argument about trends and the fact that it must be a bad coin. So...I threw it in there.
Here is what I love about this lesson: They debated. They theorized. They played. It happens this way every year, just on different levels. Today, I had to throw them out of my room at the end of class so that they could get to their next class on time. They discussed it in the halls as they walked to their lockers.
The lesson completely blew their minds. The kids who are the strict "by the book" math students had to listen to the points of view of the global thinkers who bought into the "trend", "chaos", or "diminishing returns" theories that were being thrown around. They had to argue their belief. Sure, some kids just sat back and watched the fireworks, but I didn't see a single kid bored or disinterested. By the way, it's a lot more fun if you play devil's advocate on occasion just to stoke the fire.
At the end of the period, the kids asked me which was correct. I said, "As a math teacher, I have to say 1/2 but in reality, I prefer the chaos/global approach that tails is way overdue."
Do you know that in the past I have had parents call me after that lesson angry at me that I would plant seeds like that? I asked them about the conversation that they had with their child and reveled in every word. It took very little to make them realize that they had a deep, mathematical conversation with their child and that their child was not just thinking in school but continuing that thinking throughout the day.
Anyway, if there is anyone still reading at this point (seriously doubt it), thanks for reading. As I said at the beginning, I was really jazzed about this lesson and just had to get it written. I guess it was therapeutic, if nothing else.
Hugs.

## Comments (4)Sign in or register to postSubscribe to comments via RSS

Hi, Tom,

How have you been?

So glad you posted your lesson. Great idea to include the movie clip. (I must say, not a movie that I've seen. Sounds like I might need to check it out.) Very cool, indeed! Any other movie clips that you would recommend?

Your post brought to mind two of my favorite warm-ups, both involving graphing calculators. Short on time at the moment, but will write about them soon.

Take care,

Sybrina

Father of the Bride has a cute lead in to LCM:

http://www.youtube.com/watch?v=oYIHLUxzRr8

Immediately after the clip, we discuss how many buns he ended up buying and assume that he must have bought the same amount of hot dogs. Then I ask how many pakages of 12 it would have taken to make him happy AND keep him out of jail. It's not a huge a-ha moment for them, but it is an entertaining one.

Apollo 13 has an amazing lead in to why we practice problem solving:

http://www.youtube.com/watch?v=rNDuGuerpf8

There are a couple of expletives at the 3-minute mark, so be ready on that mute button. :)

After this clip, we do a think-pad brainstorm with the following question:

What elements of effective problem solving were displayed in this clip?

You can then have them popcorn answers from their team and write them on the board.

You'd be surprised what they see.

The most important elements that you need to add if they don't:

- They dove right into the problem

- They gave a detailed explanation of the solution

Kids need to understand why we do word problems. They are 100% correct when they whine about the fact that they will probably never need to know anything about two trains leaving Boston and Chicago simultaneously. What they do need is the ability and willingness to jump into a problem and the ability to communicate their thoughts once they have solved it.

There's also the scene at the end of The Wizard of Oz when the scarecrow receives his brain. He incorrectly states the Pythagorean Theorem when touting his newly-gained intelligence.

Warm-Up #1: graphing calculator windows

Students had learned how to use calculators to graph linear functions and had been introduced to window settings. Before class, I entered the equation y = 7x into my TI-83/84 overhead calculator in the y0 slot (so the equation would not appear on the screen when I pushed the y= button). I started with a Zoom Standard window but divided x-min, x-max, and x-scl values by 7. The resulting graph looked just like y = x. I projected the graph as students entered the room and challenged students to name my equation. Students soon realized that the graph was not as it appeared and I was then asked to show my window. Guess, check, and revise led students to the correct equation. This served as a springboard for a great discussion about the importance and significance of windows.

Warm-up #2 -- domains and domain restrictions

Students were presented with the question: Are (x^2 - 9)/(x + 3) and x -3 equal? Some students went to work factoring and dividing out common factors. Others grabbed calculators and entered the first expression in y1 and the second expression in y2. Because it always seems to be the initial window of choice, students used Zoom Standard. Along with their algebraically minded classmates, they felt confident that the two expressions were indeed equal. I then stated that I disagreed and proceeded to graph the equations using an Integer window. This window allows students to see the hole of discontinuity when graphing y1. (Pressing trace and arrowing to x = -3 provides a numerical look at what is happening.) When graphing, if you change from the line format to the line with the ball, you can actually see the second equation being graphed over top the first -- and see the hole being filled in. This warm-up led to a great discussion about domains and domain restrictions, including finding restricted values algebraically.

Slope Haikus

After an extensive unit on slope, I had Algebra 1 students write a haiku about the topic. The haiku could be about any part of our unit --from data collection activities, to graphing calculator family of function explorations, to numeric and/or algebraic and/or formulaic representations. At the end of that particular year, the district held an academic night to showcase student progress/learning. Each teacher was asked to create a display. For my Algebra 1 classes, I displayed student haikus. When we put together our 60+ haikus, we really come away with an amazing picture of the meanings of linearity and slope.

Senior /Freshmen Proficiency Test Project

For several years, I taught mathematics at an alternative arts-based high school. In spite of having a population of budding artists, sculptors, musicians, and actors, our principal required everyone take 4 years of mathematics starting with Algebra 1. (Smar man -- very smart man!) Two of my groups were seniors in a college preparatory class. These were kids who had managed to make it through Algebra 2, but had no desire to take Pre-Calc or Calc. Needless to say, these kids were getting pretty tired of math class by the time mid-March/early April came around. Meanwhile, my freshmen were facing a required-for-graduation proficiency test in May. So, my college prep kids spent some time analyzing proficiency test questions. They then went to a local Science museum and wrote proficiency-test-like items using the museum exhibits and location as the questions' context. Once initial questions were written, students spent several classes editing and fine-tuning their work. A few students (the project editors)then returned to the museum to finalize questions before compiling the freshmen's final slate of questions. My freshmen then went to the museum, and in groups of 3 or 4, worked to answer the proficiency review questions. For the next few weeks in the freshmen classes, we opened class with a couple of the museum questions as a final review for the proficiency test.

This was such a win-win. Not only did my freshmen benefit from a nice change-of-pace pre-test practice, my seniors left the year really feeling good about what they had accomplished in their 4 years of math. Many of these same kids had struggled with the proficiency test themselves and now found the items to be so much easier than they were for them then. In addition, several students shared with me that this was one of the best things they had done while at the school -- they truly felt as though they were able to make a difference for the younger students. And, all kids were absolutely thrilled to go on a field trip in math class!!!!

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