I got to do a Snoopy happy dance and laugh so hard I cried. It was a week to celebrate. It had the usual ups and downs that go with being a teacher. It had the normal number of challenges and two extraordinarily long days. It included math placement testing for incoming students on Wednesday night and again on Saturday morning, plus all the scoring, data collection, and analysis that goes along with it. There were tests to give and tests to grade (grading is nobody’s favorite thing). All of those tedious things pale, though, in comparison to the chance to share a moment with a student. It’s kind of a long story, but good things come to those who wait. At least, they did for me.
As I lay out my plans for the year, I always schedule the probability unit during one of those times of the year when it feels like it takes everything you have to keep students focused. You know the times I am talking about – those last few weeks before winter break when all kids can think about is the upcoming holiday, the weeks immediately following testing when kids have had their routines turned upside down and are having a hard time readjusting to normal routines, or when spring is in the air and they can smell summer vacation right around the corner. The magic in this thinking is that students have so much fun in the probability unit that they don’t see it as work. I am lucky enough to use a really well-designed unit by Connected Math. It is full of experiments and contexts that are real and meaningful to kids.
A Quest for Sugary Cereal
I started out by introducing students to Kalvin. Kalvin was on a quest to eat Cocoa Blast cereal rather than the Health Nut Flakes that his mother wanted him to eat. Kalvin convinced his mom to let him use a coin toss each morning to decide which cereal he has for breakfast. If he got heads, he could have Cocoa Blast; if he got tails, Health Nut Flakes was the cereal of the day. Students then simulated Kalvin’s breakfast selection for the month of June by doing 30 coin tosses
Using this data, they looked for trends as they increased the number of tosses. Next, they combined their data set with their classmates to form a class data set. Using this class data set, they found the experimental probabilities of heads and of tails by finding the percentage of each. They also used the data to make predictions about Kalvin’s cereal consumption. (I could see the plotting in students’ heads begin. Would this work on their parents?)
Looking for Better Odds
Kalvin didn’t think a 50-50 outcome was good enough and so applied his intelligence to finding a better option. He considered the possibility of using a cup toss to get a better shot at a daily dose of sugar in his breakfast. He wisely realized that it might be wise to test it out before proposing it to his mom, though. Students simulated his experiment by tossing a paper cup 50 times. They then determined the fraction of the tosses that the cup lands on the side and the fraction that it lands on the end and finally convert those fractions to percentages. They used this data to decide which option that they thought Kalvin should choose.
Next, they combined data to form a class data set to make a final decision regarding Kalvin’s choice of “side” or “end” as his desired choice to get Cocoa Blast. This combining of data was a nice way for students to see the effect of the Law of Large Numbers and to consider why it is important to do many trials. They used this insight to justify why Kalvin is wrong when he decides the cup toss is no better than the coin toss after getting one “side” and one “end” on the first two days.
The lesson poses this scenario as a question of whether Kalvin’s thinking is correct. I had students form an Agree/Disagree Circle. Those who agreed with Kalvin formed a circle facing outward. Those who disagreed with Kalvin formed an outer circle facing inward. Students then debated the question with the person they are facing. I tried this because I wanted to try this out as a formative assessment structure, but wasn’t really sure if it would work. The answer seemed really obvious to me. Interestingly enough, I had one class in which one of my strongest students started out as the only one to agree with Kalvin. He argued that Kalvin was correct based on his experimental data. This student converted the majority of the class to his way of thinking. This made it really clear to me that while the class had talked about how a larger data set was a better predictor in experimental probability, students hadn’t fully grasped that they shouldn’t draw conclusions based on a few trials. They had been giving the “right” answers, but they didn’t really understand it.
The Quest Continues
Kalvin continues to look for ways to increase his consumption of Cocoa Blast. His latest thinking is that tossing two coins might lead to better odds. If the two coins match, he gets Cocoa Blast. Otherwise, he has Health Nut Flakes. Students conducted an experiment tossing a pair of coins 30 times to simulate a month of cereal consumption.
They then went on to create and use a class data set to find the probabilities. Finally, they used their data to decide if they thought a match and no match had an equal chance of occurring.
Next, students considered how many ways a “match” can occur and how many ways a “no match” can occur. Based on the outcomes, they decided if a “match” and a “no match” have the same number of chances of occurring.
In the lesson, the photo shown with the problem context showed two quarters with heads as a match and two tails as a match. The photo showed the two quarters with a head and a tail as a “no match”. I don’t know if the authors were intentional with this choice, but it definitely drew out a common misconception. I wanted to draw out this misconception in a concrete way without telling them the outcomes, so I gave each student two different denominations (a penny and a dime) for the experiment. Every time I teach this lesson, there are students who think that there are 2 outcomes for a “match” and 1 outcome for a “no match”. As we discuss this, other students will then raise the argument that a heads on the penny and a tails on the dime is not the same as a heads on the dime and a tails on the penny. I think this discussion is really powerful. Letting kids make this discovery through their discussion makes the knowledge theirs in a way that my telling them never could.
The lesson wrapped up with a scenario in which Kalvin was considering moving on to a thumbtack toss. He decided to conduct an experiment before proposing it to his mother to see which outcome he should choose. His friend got heads 6 times out of 11 and Kalvin got heads 13 times out of 50. Students had to decide which result is more reliable and justify why.
Gee Whiz, Everybody Wins!
Gee Whiz, Everybody Wins! is a game in which the player predicts what color block he or she will draw out of a bag containing red, yellow, and blue blocks. The player then withdraws a block from the bag. If the prediction is correct, he or she is a winner. The class played the game and I kept a record of the number of times each color was withdrawn. The class then calculated the experimental probability of each color.
I then showed students the contents of the bag (9 of one color, 6 of another color, 3 of another color). They found the fraction of each color and I explained that the fraction is the theoretical probability of withdrawing that color block. They found the sum of the theoretical probabilities and decided whether they would prefer to go first or last in the game.
Exploring Probabilities by Analyzing a Game
Students explored some important ideas about probability by considering different questions about a scenario in which they had a bag with 2 yellow marbles, 4 blue marbles, and 6 red marbles. A single marble was withdrawn. They began by finding the theoretical probability of red, of yellow and of blue.
Students then found the sum of the probabilities. Next, they were asked to find the probability that the marble is not blue. I love this question. Different students devise different ways to find the answer which brings out a lot of interesting discussion. Some students will add up the number of red and yellow marbles to find the answer. Some students use the idea that the probabilities add to one and just subtract the probability of blue from one. The discussion surrounding this question lets students discover the idea that P(not A)=1-P(A) for themselves.
Students then considered the probability that the marble was red or yellow. This is a great follow-up to the previous question because some of them have already used this idea to find the probability of not blue. The answer seems obvious, but it can lead to a great discussion of what it means to take the probability of one thing OR another. Students readily add the two parts and then take the part of the whole. This then raises the question of whether you could add the part of the whole and the part of the whole. This leads to the question of whether this would always work. I think it is a great way for students to figure out that P(A or B)= P(A) + P(B) if they are independent events.
The follow up to this asked students to find how many blue marbles they would need to add to the bag in order for P(blue) to equal ½. Seeing student thinking on this question is always fascinating. I think it really shows the depth of their understanding. It is great because it has multiple entry points and multiple solution paths. Some students will just add two blue marbles. There are six red marbles and P(red) is ½. They think that making blue equal to red will make it have a probability of ½ as well, failing to consider that “whole” has also grown. Some students will use a guess and check strategy, adding one blue marble and finding the new probability iteratively until they get P(blue)=1/2. Some students realize that the new total for blue will need to equal the sum of the red and yellow in order for the probability of blue to be ½. They then subtract the current number of blue from that sum. By carefully choosing who shares their thinking and in what order, I can allow the discussion to address student misconceptions and expose students to different ways to think about the same problem.
Winning the Bonus Prize
This lesson introduced the idea of finding the theoretical probability of compound events using a tree diagram. As I taught students how to make a tree diagram, I had them use different colors to highlight each path through the tree to find the different outcomes.
After showing students how to make a tree diagram, I had the class play a game in which they draw a block out of each of two bags. Each bag contains a red, a yellow, and a blue block. They predict their outcomes before drawing. They win if they correctly predict both blocks. As the class played the game, I recorded the outcomes. Students found the experimental probability of each outcome based on the class data set. They then created a tree diagram and found the theoretical probability of each outcome and compare them to the experimental probability.
Time to Do a Snoopy Happy Dance
At the end of the “Bonus Prize” lesson, Luke reached across the table and opened up the canister of Zap sticks, staring thoughtfully at the contents.
Zap is a game that I have students play from time to time to develop fluency with number operations. It’s not a brilliant game and does nothing for conceptual understanding. I use it after I have taught the concept conceptually to develop fluency in an engaging way (it is a Marzano high engagement strategy). Students play it for five minutes as a review activity at the end of class or as a warm up to get them on task and focused as soon as the bell rings. In the game, some of the sticks have problems, some of the sticks are “Take another turn”, some of the sticks are “Take a stick from another player”, and some of the sticks are “Zap – put all of your sticks back”. The winner is the player with the most sticks when I call time. Kids love the game because it is fast paced and the winner gets a piece of candy. I like it because it is an easy, fast way to get a lot of practice with boring things like rational number operations in a fun way.
On this particular day, students had played Zap as an entry activity. As Luke stared at the contents of the canister, I asked him what was up. He said, “I was just wondering what the probability of getting zapped is” at which point I was doing the Snoopy happy dance to the theme of Linus and Lucy in my mind. All of these lessons centered in a real world context that was meaningful to an 11 year old made it a natural thing to think about math in his own experiences. Naturally, my response was that we should dump out the sticks and find out the answer, which he then proceeded to do.
We then got to have a great discussion about whether that probability was constant throughout the game. How would it change as players withdrew sticks? This wasn’t a planned way to go about it, but it brought out the idea of probability when you don’t have replacement. It was real and probably more meaningful than what I had planned coming up in a few days because it was his question. I then raised the question of how things would change when someone got zapped. It was an incredibly rich discussion that all started with a student’s wondering. It might have happened anyway, but I think the wondering was a natural extension of the problem contexts in the Connected Math lessons that the students had been considering for a week.
The story about the laughter will have to wait. This post is already way too long. I will only say that I highly recommend finding room to laugh with your students when you can.