In my non-AP Calculus class, we do a unit on probability. At the start of class we watched the TED-Ed video called "The Last Banana". The video describes a two person game where each person rolls a fair 6-sided die. Player 1 wins if the highest roll on any die is a 1, 2, 3 or 4. Player 2 wins if the highest roll on any die is 5 or 6. I paused the video at the 37 second mark and at first I was just going to have my students discuss the answer to the question, "Who do you want to be - player 1 or player 2?" (Note: At first glance, it seems as if player 1 should win more often. After all, the probability of rolling a 1, 2, 3 or 4 is 4/6 and the probability of rolling a 5 or 6 is 2/6.)

But then, I deviated from my original plan and gave each student a die, pairing them up to play the game. They kept track of their wins. Our results: player 1 had 60 wins and player 2 had 69 wins. The students said that they still thought it could be a fair game, meaning in the long run each person would have an equal chance of winning. Rather than roll the die more times, students looked at pairs of outcomes together and decided who would win for each outcome. The theoretical probabilities were: 16/36 for player 1 and 20/36 for player 2. Finally, we finished watching the TED-Ed video, which summarized the concepts for the lesson.

Whenever I can give students a hands-on experience with probability, it makes the concept more memorable. I hope this one will stick with them. Plus...there is the added bonus of looking at this video again in AP Statistics from an inferential point of view. There we can ask the following question: "Assuming player 2 has a 20/36 chance of winning, are we likely to get results at least as extreme as 60 vs 69 by chance alone?"

## Thursday, August 31, 2017

## Wednesday, August 30, 2017

### Teach 180: The Value of Homework (Day 3)

What is the purpose of homework? If you asked students, some would say, "it is something we are expected to do to prepare for college." Others might say, "It's something I need to do, because it is worth 15% of my grade." And finally, a few might reply, "It is something my parents want me to do, but I have better things to do with my time." In my mind, these students don't understand the purpose of homework or at least my personal view of the purpose of homework.

I see homework as a chance for students to work with the concepts learned in class some more. Ideally, they should have learned enough that they can do many of the problems right, but they may get stuck or get some wrong, too. Getting stuck and making mistakes is part of the learning process. It isn't evil or a sign of bad teaching or a sign that the student wasn't paying attention in class. In my classroom, I have a Postulate of Learning that states "Choosing to not do an assignment robs you of an opportunity to learn." However, each year there are students that don't believe that postulate is true and rob themselves of learning by doing things like erasing a page number from the top of the assignment and putting the page number at the top for what was due for that day. This gives the illusion that they completed their homework. Or they will copy off of another student or they will use the website called slader to get all the answers to their homework.

Today I was curious if our new book for Calculus was on slader and I was happy to see the following:

Of course students could still see slader on their phone or at home, but calling it plagiarism is correct. Copying the answers and work from slader means the student didn't do or create the work himself or herself. Does this match with the definition of plagiarism? According to google dictionary, I would say "Yes".

It should be noted that ironically enough, slader has an Academic Integrity section in its FAQs. They ask their users to report any solutions that they suspect have been stolen or copied from another source. I will likely be blogging about homework in my Teach 180 blog several times this year. Stay tuned!

## Tuesday, August 29, 2017

### Teach 180: A Banged Up Knee (Day 2)

My second day of classes went really well and then the unexpected happened. Two unexpected things, actually. First, we did a simulation in Probability and Statistics. The scenario: an airline company needs to select 8 pilots at random from a group of 15 male and 10 female pilots. The results obtained by the airline were 5 females and 3 males. But wait, shouldn't we have more males than females? That is what one might think based on the ratio of the original group. Is it likely to get these results (5 females and 3 males) by chance alone? We used playing cards to simulate the results and the dotplot here shows our collective results.

Note the one dot at 8. That happened in 1 out of 65 simulated trials. But is it really that likely? Is there a 1.5% chance of this happening? We conducted a simulation in Fathom and never got 8 in our 2000 simulated trials. Next, we calculated the theoretical probability and found it to be about 4 out of 100,000. It's not that getting 8 is impossible, but it is extremely, extremely, extremely, extremely, extremely unlikely.

So, what was the second unexpected thing? If you read the title to this blog, you guessed it. Two minutes after the school day ended, I slipped in some water that someone had spilled in the hall. I came crashing down on my right knee in the process. At this moment, my knee hurts to bend it and it is being iced. A special thanks to my advisees in the hallway, Annie and Elliot, for making sure I was ok.

Note the one dot at 8. That happened in 1 out of 65 simulated trials. But is it really that likely? Is there a 1.5% chance of this happening? We conducted a simulation in Fathom and never got 8 in our 2000 simulated trials. Next, we calculated the theoretical probability and found it to be about 4 out of 100,000. It's not that getting 8 is impossible, but it is extremely, extremely, extremely, extremely, extremely unlikely.

So, what was the second unexpected thing? If you read the title to this blog, you guessed it. Two minutes after the school day ended, I slipped in some water that someone had spilled in the hall. I came crashing down on my right knee in the process. At this moment, my knee hurts to bend it and it is being iced. A special thanks to my advisees in the hallway, Annie and Elliot, for making sure I was ok.

## Monday, August 28, 2017

### Teach 180: Amplifying Student Efforts (Day 1)

Although my school has less than 180 days in our school year, I have decided to try a "Teach 180" blog this year. In the past, I have written blogs that are long and that I agonize over. (It can easily take me 2-3 hours to write one blog post.) This year I will write shorter blogs and with that there will hopefully be less agony.

Today was the first day of school and I could talk about my classes and how they went. Instead, I am going to focus on something that I read as I was registering for Math Madness. (This is an online math competition that some of my students have participated in for the past four years.) The Math Madness website lists the following as its pedagogical foundation:

Math Madness has a structure that allows students to work toward improving in problem solving by beating their own personal best. This made me wonder, "Do I have a structure that allows students to work toward improving in my classroom?" In the past, I would put a sticker on a "A" quiz or test. However, for many students a "C" or a "B" is their "A" or personal best. This doesn't mean that in my room "everyone will get a trophy" for simply showing up. What is does mean is that I need to do a better job at recognizing not just the "A" students, but also recognizing those students who work with determination to "amplify their efforts" and ultimately, become more successful.

Today was the first day of school and I could talk about my classes and how they went. Instead, I am going to focus on something that I read as I was registering for Math Madness. (This is an online math competition that some of my students have participated in for the past four years.) The Math Madness website lists the following as its pedagogical foundation:

*The idea behind Math Madness is to create a structure that motivates students throughout the entirety of a school semester, one of enough time duration and frequency that students can observe actual progress, and in turn, amplify their efforts to continue on that path.*Math Madness has a structure that allows students to work toward improving in problem solving by beating their own personal best. This made me wonder, "Do I have a structure that allows students to work toward improving in my classroom?" In the past, I would put a sticker on a "A" quiz or test. However, for many students a "C" or a "B" is their "A" or personal best. This doesn't mean that in my room "everyone will get a trophy" for simply showing up. What is does mean is that I need to do a better job at recognizing not just the "A" students, but also recognizing those students who work with determination to "amplify their efforts" and ultimately, become more successful.

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