Friday, March 27, 2015

Teaching Probability Day 8 - Conditional Probability and Two-Way Tables

Today was a great day. My fourth period is split by lunch, so today I used that to my benefit. During the first half of class, we discussed conditional probability in two-way tables. I gave students a half-sheet for notes and displayed the same tables on the Smart Board. That went well, and students caught on quickly.

After lunch, we played "Ghosts in the Graveyard" from Kim Hughey. See her original post HERE. Instead of using ghosts and graveyards, we used May flowers and April showers. The kids started out having a little difficulty with the problems, but after prompting them a little, they got the hang of it and loved the game. One student even said "These are hard, but I get it". Yes!

Today was a good day AND it's Friday!

Materials are attached.

Thursday, March 26, 2015

Teaching Probability Day 7 - Conditional Probability

Conditional Probability is probably the most difficult probability to teach. Kids don't get it easily - especially conceptually. Today is the first time I've taught a group of kids conditional probability and I felt strongly that they REALLY got it. By the end of the lesson, the kids were saying "This isn't that bad", which is in stark contrast to what I usually get.

I started by showing two LearnZillion videos to teach them about Conditional Probability. Now, to be fair, I made these two LearnZillion videos, so I actually used the slide download and went through it piece-by-piece with them. If you choose to use the videos, you need really good speakers because the video is really quiet.

Regardless, we went through the two videos, and I made the kids a notesheet to accompany both videos with key takeaways. They filled them out and we heavily discussed the conceptual part of conditional probability... what are we REALLY doing? At the end of class, I pulled a couple of example problems from a worksheet I had.

This is the most success I've ever had with teaching conditional probability. Tomorrow, we are going to talk about conditional probability in two-way tables and do some practice.

LearnZillion Conditional Probability Video #1: https://learnzillion.com/lessons/2446-understand-conditional-probability-using-scenarios

LearnZillion Conditional Probability Video #2: https://learnzillion.com/lessons/2447-calculate-conditional-probabilities

Teaching Probability Day 6 - Multiplication Rule Practice "The Grudge"

Day 6 was just a lot of us practicing different kinds of multiplication rule problems. I didn't want to do the traditional worksheet, so I decided to try a new game I heard of called "The Grudge". The Grudge is from @nathankraft1 on Twitter. To see his post on how to play "The Grudge", click HERE. I've heard from a few people that this game is great, but it didn't really work for me. Maybe it is the word problems that are the issue. I'm not sure. The first time around, my kids played individually. That was time consuming and they could knock each other out in about 5 min. I wanted to make the game more robust, so we changed the rules the second go-around. My kids assembled into five teams, and instead of just losing x's, they could also put an x back for their team, so they'll still have a chance of winning. I had a good group of kids during this class, so they kept giving me suggestions on how to change the rules to make it more fun and reasonable, but we never came up with anything great.

Maybe somebody has a suggestion for me?: What am I doing wrong?

Teaching Probability Day 5 - Multiplication Rule

On Day 5 of our probability unit, our objective was to learn the multiplication rule. This lesson heavily focuses on the multiplication rules and a deck of cards. Our end-of-year test seems to have heavy focus on cards, especially for the multiplication rule, so it's an obvious choice. My only concern is that my kids don't know much at all about a deck of cards, so it's sometimes hard to distinguish if errors are due to multiplication rule misunderstandings or their lack of knowledge concerning a standard deck of cards. Anyway, materials for this lesson follow.

Wednesday, March 25, 2015

Teaching Probability Day 4 - Addition Rule Practice

We're approaching the middle of our ten day probability unit in my standardized test prep course. The students discovered and applied the addition rule on Day 3, so Day 4 needs to be addition rule practice. I wanted to come up with a bit more interesting way to practice these problems other than a nice black and white worksheet. I decided to make turnover cards for my kids. I learned about turnover cards at KCMC '15. Vonda Stamm of Making Math Magic did a presentation about math games. I LOVED them all. However, turnover cards really lend themselves to lots of different content. I made two sets of turnover cards for the addition rule. They really work just like the around the world activity, but they are easier for word problems, since kids don't have to squint or climb over people to read the whole problem. Kids pick up the "Start" card, work the problem on the back, and then identify the next cards with the correct solution on the front. By the end of the turnover cards, the final solution, when turned over, should read "End".

I'm attaching both sets of turnover cards I made for the addition rule. Be aware, though, that a few of these ask kids to create two way tables before doing an addition rule calculation.

Set 1 Back

Set 2 Back

Sunday, March 22, 2015

Teaching Probability Day 3 - Addition Rule (One of My Favorite Lessons Ever!)

Having taught the "Addition Rule" for three years now to all of our Algebra II students, I knew that my kids always got bogged down with the notation and formulas. So, I threw that out and decided to teach it without the formulas this year (at least to start). When kids entered the door, they received a eye card and a pet card (pet cards are from lesson 2). We started talking about the probability that a randomly selected student had a blue or green eye. I asked students who had blue eyes to hold up their cards and wrote the probability (in the set I'm providing it's 6/20). Then, I asked for students who had green eyes to hold them up and wrote down the probability (in the set it's 2/20).  We added the fractions and got 8/20. To check our answer, I asked all students who had a blue or green eye to hold them up again. The kids saw that the answer was indeed 8/20.

Then, we picked up our pet cards. I asked kids what the probability was that a randomly selected student has a dog or female pet. The key here is to make sure the deck is stacked in such a way that yields a highly suspicious probability. (For this first example, we want less blue cats and more pink dogs) We went through the same exercise. I asked students who had dogs to hold up their cards. We counted and wrote that probability on the board/notes. This varies, since I don't use all the cards in the deck (my classes are usually a couple of students smaller than the 24 in the deck). Then, I asked students who had female "pink" pets to hold up their cards, counted them, wrote the probability on the board/notes. I always choose enough pink dogs and few enough blue cats so that the probability is suspicious - either 1 or over 1 is best to make kids suspicious. I've always had a key say... "Hey... That can't be right." We have a discussion about why that can't be right (probability distributions add to 1, or - if you got 100% probability - that everybody in the room would have to have a female pet or dog. I then point out the student in the corner with the blue cat. Regardless, I turn to a student with a pink dog and ask how many times they raised their hand/were counted. They understand that they were double counted, and thus we need to subtract the probability of the intersection - or, to them - the probability of pink cats. Kids with pink cats hold up their cards, we count, and subtract that probability on the board/notes. We get a final answer. Then, I ask kids who have a female pet or a dog to hold up their card and count to check our work. It works beautifully and kids don't struggle. They get it! My favorite moment is seeing the light bulbs during the "pink dog" exchange.

I've tried to do this activity before with kids' shirt colors or hair colors; it just doesn't work as well for me.

I'm attaching my materials here for your use. If you choose to use them in your class, please drop me a quick note of how it went, what you'd change, etc. I'd appreciate the feedback.

Deck of Animal Cards (Click on Image for Google Drive File):

 Deck of Animal Cards for Two-Way Tables and Addition Rule.docx

* One of these questions prompts kids to use their "Playing Card Crib Sheet" because my kids don't know cards well at all. I just give them a printout from Wikipedia.

Teaching Probability Day 2 - Two-Way Tables

On day 2 of my 10 day probability unit, I introduced my kids to two-way tables - contingency tables - relative frequency tables - whatever you want to call them. I introduced this through a shape/color sort on the Smart Notebook file as well as assigning each student a gender/species pet card. When I was presenting this lesson to a group of teachers this weekend, they brought up that it wasn't good to depict gender with a stereotypical pink/blue color. I was just trying to make the exercise really visual from afar. If you have a better idea for this or make a different set of cards, let me know, and I'll add them here. While the cards might be a little overkill for this lesson, they get reused on day 3 (I assign students the same card for both days).

The files for Day 2 are available for download below. Let me know if you have any suggestions or, if you use them, how this works for you in class! By the way, this is a very basic introduction to relative frequency tables. The use of these tables will be spiraled through the introduction of the Addition Rule, Multiplication Rule, and Conditional Probability.

Student Notes Sheet (Click on Image for Google Drive File):

 Two-Way Tables Student Notes Sheet.docx
 Two-Way Tables Smart Notebook File
 Pet Cards File for Two-Way Tables and Addition Rule.docx

 Practice Problems for Two-Way Tables.docx

Teaching Probability Day 1 - Sample Spaces and Fundamental Counting Principle

In preparation for our EOCs, I'm teaching probability to our Algebra II kids. This is the third year I've taught this unit, and I've chosen to completely re-work it for this year and make it much more conceptual. The previous two years, I've taught it very much in keeping with the way I teach probability rules in AP Statistics, but mastery just wasn't where I wanted it. So... I'm going to upload my newly-created resources for other folks to use or help me modify to make better. I should be clear that I'm not taking an incredibly formal approach to probability as in years past. We're tested on QualityCore (from ACT). I've looked at the sample questions from ACT, and it's just not warranted. The Multiplication Rule is simply with or without replacement - kids don't actually have to know the formula with the conditional probability within it. And, as for conditional probability, kids need to simply know the formula and how to calculate it from a two-way table.

Anyway - Day 1 is simply sample spaces and the fundamental counting principle. It went really well (this lesson usually does, though - it's easy!).

I give the kids a note sheet and rely primarily on my Smart Notebook Interactive File. Files are included below. If you use them, please tweet me or leave a comment to tell me how it goes!

Note Sheet (Click on Image for Google Drive File):
 Sample Space and Fundamental Counting Principle Notes Sheet.docx

* By the way, you can delete the last slide or make up values for the year and number of tags issued. I've tried to called the Kentucky Transportation Cabinet for numbers multiple times and can't get anyone to answer.
 Sample Space and Fundamental Counting Principle Smart Notebook File
Practice Problems (Click on Image for Google Drive File):
 Sample Space and Fundamental Counting Principle Practice Problems.docx

Wednesday, March 18, 2015

Breakthrough!

I've been having a rough year. Things haven't necessarily gone the way I would have hoped in many of my classes this year. There is a disconnect between my normal teaching strategies and the way my sophomores learn. It's been a struggle, but I'm still fighting to figure it out. Today, however, was a great day.

In my sixth period geometry class, we began with a discovery activity about triangle midsegments. It's my favorite lesson of the year. I think it's because it spirals so many of the prior ideas we've learned. Anyway... I asked "How many midpoints does a triangle have?" and thus "How many midsegments does a triangle have?". I know - I know.. this is one of those crappy questions with a definite answer. Bad teaching moment. However, one of my students turned it into a definite win. He raised his hand and said infinite. Now... he is THAT kid. You know - the one that sits in the corner and waits until he can chime in with something "cool" like infinite or no solution or Illuminati. But, this kid is brilliant - just too cool for school. He went on to say that the three midsegments of the triangle form a second triangle, which also will have three midsegments, etc. Brilliant - and yes- just yes!

Student B. Says... "So tell me again why we can't find the midpoint of a line?". Other kids respond that to take a midpoint we must know where the segment begins and ends and that we don't know this about a line. So, student B says can we just use the midpoint formula and say negative infinity + positive infinity divided by 2. We then discussed that we can't treat infinities like variables and that a negative infinity and positive infinity don't necessarily reduce to zero, unless we assume equal infinities. We started talking about how some infinities could be more infinite than other infinities... awesome conversation.

Student C then says "Doesn't a line segment have infinite midpoints?". I didn't really get where he was going, so I asked him to explain. He said "You can take the midpoint of the line segment, then take the midpoint of that point". Another student chimes in "The midpoint is a point. You can't find the midpoint of a point." So, attending to precision, he re-phrases and says "Can't you take the midpoint of a line segment then take the midpoint of one half of the line segment, then take the midpoint of that, etc". I told him that what he was doing was partitioning the segment into certain ratios - halfs, fourths, eighths, sixteenths, etc.... all while the newest midpoint was approaching the first endpoint (according to how he'd explained it).

All three of these awesome conversations from this one discovery activity about triangle midsegments. Here's the link: http://wiki.mhshs.org/images/f/f8/Midsegment_of_a_triangle_theorem.pdf

Now... if we could only have these types of discussions everyday.