My son and I were looking through an issue of Fun to Learn Friends magazine recently and we ran across a game called "Spring Bingo". He was quite interested and wanted to give it a try. He is in grade primary and can subitize the pips on a six sided die as well as confidently sum the values on two dice. Just the skills we need to play this game.
We played several times. At the end of most of the games, my son got frustrated trying to roll either a 2 or a 12. He doesn't quite understand why it takes so long to roll one of these numbers. George's bingo card has both of these numbers out of the eight numbers on the card. This seems unnecessarily cruel. Ted's card has the 2 but not the 12. This made me wonder if this is a fair game or if one of the cards has a better probability of winning. Time for some math...
It appears that Card B is slightly better than Card A. I'm not sure if this would make a significant difference in the outcome of the game (i.e. rolling all the numbers on your card before your opponent does). I wonder what the average number of rolls it takes to complete each bingo card is? How much of an advantage does going first in this game give? If the player with Bingo Card A goes first does this equalize the advantage of the better Bingo Card B? These would be great questions for the Mathematics 12 research project outcome (MRPO1).
To try to answer some of these questions, I thought that writing some code would be helpful. I know a grade 8 student that completed the Introduction and Intermediate Programming with Python courses from Art of Problem Solving. I contacted him and he graciously created a very nice Python program to simulate this game for me. I modified the code a bit so that it just plays one bingo card and counts how many rolls it takes to complete the card. The average number of rolls in 100,000 games for "George's Card" was 58. The average number of rolls in 100,000 games for "Ted's Card" was 48. I was a bit surprised that there was this much difference. Playing with Ted's bingo card appears to be an advantage.
Next I modified the code again so that it plays the game with George going first to count how often George wins. In 100,000 games, when George went first, he won 50703 of the games. I again modified the code so that Ted plays first. In 100,000 games, when Ted went first, he won 50806 of the games. It seems that going first is an even greater advantage than having the better bingo card.
What I really like about this game is that there are mathematical outcomes that can be addressed with this activity across a wide range of grade levels. At younger ages, students are practicing subitizing and adding numbers. One variation of this game is to play it solo. This might be a nice option for a math station. I found several examples of "Roll and Cover" games where students have a sheet of paper filled with numbers (from 1 to 6 or from 2 to 12) and students roll the die or dice and cover the number (with a token or a bingo dauber) when they roll it. Just do a Google search for "roll and cover math game" and you'll find lots of examples posted online.
For students looking for an opportunity for enrichment, they can make variations of this game. They could also write computer code to simulate this game (using Scratch or Python or some other language). They could also do some statistical analysis of the game to see how fair it is. There are so many options with this simple game.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 1 N02 - Students will be expected to recognize, at a glance, and name the quantity represented by familiar arrangements of 1 to 10 objects or dots.
Mathematics 1 N09 - Students will be expected to demonstrate an understanding of the addition of two single-digit numbers and the corresponding subtraction, concretely, pictorially, and symbolically in join, separate, equalize/compare, and part-part-whole situations.
Mathematics 2 N10 - Students will be expected to apply mental mathematics strategies to quickly recall basic addition facts to 18 and determine related subtraction facts.
Mathematics 5 SP04 - Students will be expected to compare the likelihood of two possible outcomes occurring, using words such as less likely, equally likely, or more likely.
Mathematics 6 SP04 - Students will be expected to demonstrate an understanding of probability by: identifying all possible outcomes of a probability experiment; differentiating between experimental and theoretical probability; determining the theoretical probability of outcomes in a probability experiment; determining the experimental probability of outcomes in a probability experiment; comparing experimental results with the theoretical probability for an experiment.
Mathematics 7 SP06 - Students will be expected to conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or other graphic organizer) and experimental probability of two independent events.
Mathematics 8 SP02 - Students will be expected to solve problems involving the probability of independent events.
Mathematics 10 Essentials G1 - Express probabilities of simple events as the number of favourable outcomes divided by the total number of outcomes
Mathematics 12 P03 - Solve problems that involve the probability of two events.
Mathematics 12 MRP01 - Research and give a presentation on topic that involves the application of mathematics.
I know many teachers who say they wish they could go to a math conference but they are not sure of the logistics of applying for funding, applying for leave, getting sub coverage, etc… I have gone to many conferences over the past 10 years and always return feeling rejuvenated and excited to do my job!
I have attended the NCTM Annual Conference although I may try a regional conference in the future. The following is based on my own experience; please review your school board and Article 60 guidelines before making plans.
Things to consider before booking your conference:
So now you have decided the conference, the city and travelling companions, here are some of the logistics of applying for funding:
Once you have gotten approval for leave and have confirmation on how much you will be reimbursed in expenses, it’s time to start planning your trip! I often scout out hotels and flights even before I have completed the steps above. My preference is to stay at one of the conference affiliated hotels. There are complimentary shuttle busses that run from the hotels to the conference centres all day. This is a big bonus in my eyes. Sometimes, I have used the shuttle busses just to get to another part of the city! You can find all the information about accommodations on the NCTM website. The best hotels (location or price) usually book up early. If I don’t have all the proper approval from my principal yet, I still reserve my hotel room. I can always cancel the reservation if plans fall through (unlike conference registration and plane tickets. Wait to book those!).
Take note of the early bird registration dates for the conference. It will save you about $50 if you register before a certain date. If you know of a bunch of people who are attending the same conference, investigate group rates. Also, I usually scour the internet for NCTM conference registration coupon codes. I once saved myself fifty dollars by using a random code I found online.
So now that you have gotten approval for leave and funding, registered for conference, reserved your room and booked your plane ticket, you need to plan your time at the conference. With thousands of sessions to choose from, selection can feel like a nightmare (similar to those many of us get in the days before a new school year begins). For this reason, you want to choose wisely. Like the first days of school, this planning you do in advance will certainly impact the days ahead.
More on strategies to navigate the conference booklet and choosing your sessions coming soon!
For anyone interested in data visualization, the r/dataisbeautiful subreddit is an amazing place to explore (you can also follow @DataIsBeautiful on Twitter). This subreddit has an active community of posters who create and discuss a wide variety of data visualizations. Shortly after the first successful landing by SpaceX of a reused Falcon 9 rocket, an incredible feat of engineering, the visualization below was created and shared by reddit user Brenden2016.
Even more interesting than the visualization was the discussion that followed. Over 200 comments (as I write this) have been posted regarding this visualization and suggestions of how to improve its powerful story of the increased efficiency and success of SpaceX Falcon 9 landings. Several issues emerged:
Additionally, two users created and posted alternate versions of the graph. One user, nicocote, created a double bar graph using additional launch data that didn't appear in the original. Another user, azura26, created a step graph instead of a line graph.
The full list of Falcon 9 landings can be found on the list of launches on Wikipedia. I took this data, summarized it and put it into a Google Sheets document to share (https://goo.gl/Ihr2Io). Please feel free to use it to generate your own data visualization or use it with students.
In the Classroom
This set of data appears to be a nice one to use in a classroom. It is a fairly small data set, is current and from the real world and (for me at least) is an engaging and interesting story. In a grade 8 classroom working on data presentation, I might start by showing the students some dramatic video of Falcon 9 boosters both crashing and landing successfully. Fiery explosions always make an impression! Then we could look at the data showing how SpaceX has done an impressive job of learning from their mistakes (and perhaps how this relates to a growth mindset in the mathematics classroom). We could then look at the data visualization from above and brainstorm, as a class, what features of the graph are positive and which could be improved. We could also answer the question, "what is the story this graph is trying to tell?" We could then break into groups, review that data and see if the groups could come up with a different way of telling that story using the data. Perhaps a different type of visualization or even an infographic could be created.
What would you do with this data in your classroom? Please let me know.
Nova Scotia Mathematics Curriculum Outcomes
Grade 8 SP01 - Students will be expected to critique ways in which data is presented.
Mathematics Essentials 11 F2 - Select an effective data display for a given set of data and explain the reasons for the choice.
Mathematics at Work 11 S01 - Students will be expected to solve problems that involve creating and interpreting graphs, including bar graphs, histograms, line graphs, and circle graphs.