Below is a simple probability game that I've played with students for many years in a number of different courses. I don't really remember where I got it from but if you've seen it before, I'd love to know where it originated. I've found that students enjoy it and they really develop some good ideas about probability. Its similar to the Two-Dice Sum Game from Marilyn Burns but with some additional mental math for older students.
This is a game for two or more players. Each player creates their own score card by drawing two concentric circles and then divide it into eight sectors. Choose eight different numbers between 2 and 12 and place them in the sectors in the outer ring. In the inner ring of the circle place numbers that add up to 100 (it is easiest to stick with multiples of 5). Roll two dice and add the numbers together, if the total is one of the numbers in the outer ring of your score card, you score that numbers value (the number in the inner ring). Players alternate rolling and scoring on their card until one of the players has 150 or more points (or to 100 for a bit shorter game). A google doc handout for this is available here.
Sample Score Card
Typically, the first time students play, they will randomly select numbers and values for each sector. After students have had the chance to play a few games, stop and ask them to look at different score cards to see if there any common characteristics of winning cards. What makes a good score card? Is it all just luck or are some cards better than others?
Developing a Strategy
What makes a good score card? Here are some common student observations:
You might ask students to roll two dice a bunch of times and record which numbers come up each time. Create a bit dot plot or a bar graph at the front of the classroom using all of this data. Ask students to predict what the shape of this graph will be. You could also use an online tool to model lots of dice rolls to see how it compares to the class data. You could then discuss the theoretical probability for each sum and compare it to the data you gathered.
I always find this a fun activity and a nice way to start a discussion about probability and strategy. A nice question to start the class on the following day would be a Would You Rather? math prompt, "Would you rather flip 2 coins and win if they match OR roll 2 dice and win if they don't match?" Another great follow up activity would be Don Steward's Dice Bingo.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics at Work 12 P01 - Students will be expected to analyze and interpret problems that involve probability.
Mathematics Essentials 10 G4 - Compare predicted and experimental results for familiar situations involving chance, using technology to extend the number of experimental trials.
Mathematics 8 SP02 - Students will be expected to solve problems involving the probability of independent events.
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.
About a year ago, I signed up for Samira Mian's Udemy course on Islamic Geometry. I also purchased a copy of Eric Broug's book Islamic Geometric Patterns. I wanted to learn the basics so that I could determine if this might be a good way to satisfy the grade 7 mathematics geometric constructions outcome. I designed a short unit that I described last year. Recently, I decided to try replicating some of these patterns using the online Desmos calculator and geometry tool. I think having some experience drawing these patterns with a compass and straight edge was helpful.
If you're looking for some Islamic geometric patterns to try, YouTube is a great place to get some ideas. There are some great instructional videos from Samira Mian and Nora Youssef, among others. The first pattern that I tried was a Star and Hexagon pattern that I learned from Samira's Udemy course. I learned that sticking with exact values are worth the effort. Rounding intersection points and slopes of lines to the nearest tenths or hundredths place work well at first but the errors compound and things start to get messy down the road. Interlacing the pattern gave me lots of practice with domain and range restrictions.
8 Fold Rosette
Nora Youssef has a nice video tutorial on for drawing an 8-Fold Rosette pattern. I did this pattern twice. The first time I constructed the basic pattern and the second time I added interlacing. I used the polygon function to add colour and figured out how to use trigonometry to rotate the polygons around the origin. This made it really efficient. I created a table with the vertices of the polygon and then just duplicated and rotated that polygon around the rosette. I duplicated the polygons multiple times to make the colours bold.
You can see from my notebook below that some of the math took me a few tries (this goes on for several pages). To make the weave for the 8 fold rosette, I made lines parallel to the original with a distance of 0.5 above and below. Each ribbon was then 1 unit wide. I was working with the equations in point-slope form. I'm pretty sure that there are more efficient ways to do these calculations but I haven't discovered them yet. I really like how these messy bits encourage me look for more efficient and elegant methods.
Desmos Geometry Tool
After working with the Desmos calculator for a while, I wanted to give the geometry tool a try. I decided to try a pattern that I saw on the Pattern In Islamic Art website. This site has some great resources. The pattern that I tried was from David Wade's book Pattern in Islamic Art. The geometry tool requires much less algebraic manipulation, but I find hiding the underlying grid is much more tedious than in the calculator. Everything has to be hidden individually instead of turning a whole folder on or off in the calculator. I've drawn this pattern in the past by hand and it would have been much more difficult if I didn't have that previous experience.
I've tried tiling some designs to cover the plane but I haven't come up with any good methods for this yet. I've also tried using sliders to dynamically adjust some of the relationships between the sizes of the pieces in these designs. These are great challenges and are helping me learn new features of Desmos. Dan Meyer wrote "If Math Is The Aspirin, Then How Do You Create The Headache?" I hesitate to call these graphing projects "headaches" because I enjoy the challenge. Regardless, this is a case where my need for mathematical solutions guide my learning and give me reasons to explore new graphing methods.
I recently had the pleasure of attending the Nation Council of Teachers of Mathematics (NCTM) Annual Meeting and Exhibition in Washington, DC. This was my first big math conference. I attended some great presentations. One message that I heard reiterated in a number of presentations and one that resonated with me was about rich tasks. The message was that having a rich task is only a starting point for effective mathematics instruction. A rich task in math class is like being dealt a great hand of cards in Poker. It makes winning easier but it still takes a seasoned player with solid understanding of the complexities of the game to win a big pot.
Full Stack Lessons
Dan Meyer presented a session called "Why Good Activities Go Bad" in which he discussed a math task called Barbie Bungee. After giving summary of the task he interviewed three different teachers on their use of the task and their students' experiences. Dan asked us to think about what makes a task engaging and productive and what might make it fall flat. He talked about "full stack" lessons and how the mathematical task itself is just one component of a fully developed and presented lesson.
Sara VanDerWerf, in her presentation ‘Engaging Students in Seeing Structure’, talked about her overarching goals when lesson planning. Using routines such as Notice and Wonder and Stand and Talks, Sara supports her students to see and talk about math concepts before they are formalized. Students have a chance to engage in the mathematics and build conceptual understanding.
These types of routines which allow students to be curious about math and ask questions are important elements for getting the most out of a rich math task.
Super (Secret) Mathematics of Game Shows
One of the most engaging sessions that I went to at NCTM was presented by Bowen Kerins. He had a fun presentation called Super (Secret) Mathematics of Game Shows. A few elements of his presentation that stand out:
Shortly after returning from NCTM, I saw these tweets above from Fawn Nguyen and from Cathy Marks Krpan. These thoughts sound like they reflect many of the messages that I heard in Washington, DC this year. This morning, I also saw John Rowe's blog post "The Secret Sauce of Great Lessons." It looks like he attended several of the same sessions that I did and had a similar reflection.
My Lesson Planning Challenge
One the first slide of my presentations, down in the notes, I often write, "The essential components of a presentation: a clear focal point, a strong flow and structure, a beautiful design and a compelling delivery." (I picked this quote up here). It serves as a reminder to stay focused and think about the structure of my presentation (or blog post for that matter). I need to create a similar reminder for planning lessons to focus on more than the task itself. I need to consider how that task will be implemented to make it as engaging and productive for students as possible.