Auctions can be a fun and engaging activity in math class. The first auction I ever tried with a class was a unique bid auction that I learned about from Dan Meyer. I called it a "Lone Wolf" auction after seeing a discussion about this type of auction from Shawn Cornally. Students really enjoyed it and it generated some great discussions about strategies. We collected some data and were able to look at some displays to analyze the game. I haven't used many auctions as part of a mathematics activity. I was always concerned that too much time would be taken up by the logistics of the auction and there would not be enough time devoted to mathematical analysis. I learned some strategies for focusing on mathematical reasoning by reading how other teachers have facilitated auctions to make engaging math activities. Here are a few below: Function AuctionSarah Carter created a function auction activity for students to deepen their understanding of what makes a relation a function. Students bid on lots containing a variety of relations... some functions and some not. The team with the most functions at the end of the auction wins. Students work as a team to try to identify which lots contain the most functions and what a reasonable price for each lot might be given their starting funds. Sarah found this to be more engaging than a traditional function/not-function worksheet. The activity generates a lot of student conversation and discussion about functions. Dice AuctionNat Banting created a dice auction activity to get students thinking about the probabilities of outcomes when rolling two dice (this activity was won the 2020 Rosenthal Prize). In his description of the activity, Nat makes sure to include plenty of time for student teams to discuss their strategy and to try to assign a reasonable price to each event. He also builds in some reflection questions for students to answer at the end of the activity. This activity reminds me somewhat of the Borel Dice board game. Definite Integral AuctionI decided that an auction might be an engaging way for students to practice evaluating definite integrals in calculus class and so I created a definite integral auction activity. I incorporated some of the ideas from both Sarah's and Nat's auction activities. My auction is described below. As in Sarah's activity, I included time for students to talk and strategize by starting with five minutes to review the "auction catalogue". This page shows all the values that will be auctioned. As in Nat's auction, I scheduled in time at the halfway point for students to assess their situation and review the remaining lots. This gives students time to calculate their current integral and determine how they might increase its value even further. I also included some of Nat's reflection questions at the end to consolidate the activity. I got some additional ideas from Lola Morales (@lolamenting) when she posted on Twitter how she was going to use this activity in her classroom. If you have any tips or suggestions about auctions in math class, I'd love to hear about how you facilitate these types of activities. EL
Being on Twitter and following hashtags like #MTBoS and #ITeachMath allows me to see classroom mathematics well beyond my physical horizons. I get to glimpse creative and engaging mathematics education around the globe. Recently I saw a couple of different ideas that I've tried to adapt and apply for myself. Mysteries
Since the Nova Scotia grade 8 classes are working on integer multiplication and division, I decided to create a math mystery of my own. Another nice source of math mysteries is the book Mathematical Team Games: Enjoyable Activities to Enhance the Curriculum by Vivien Lucas. Treasure Hunt
I liked this idea because it is relatively easy to create; just a find a series of questions with unique answers. Also, students get instant feedback. If their answer isn't on the map, they know they've made a mistake. I would call this purposeful practice as there is a goal to achieve at the end of the activity. There is a reason to persevere. Once students are familiar with the activity, you could give them a blank template (or they could hand draw their own version) and they could work in small groups to make their own treasure hunt activity (and answer key) and share it with each other. The Role of PracticeI recently read Mark Chubb's (@MarkChubb3) blog post on the role of practice in math class. He discussed the differences between "rote practice" and "dynamic practice". Rote practice involves following procedures, drill and repetition while dynamic practice involves active student thinking, playful experiences and puzzles. I think that the Mystery activity is a more "dynamic" activity than doing the Treasure Hunt activity. However, I think that creating your own Treasure Hunt activity does involve additional characteristics of dynamic practice. EL
Practice is important. Whether it is playing the piano, shooting free-throws, popping an ollie or solving a quadratic equation you need to practice to improve. Some practice routines are more effective than others at helping students solidify their understanding. Practice can often seem tedious and it can be difficult to maintain the motivation to practice. In mathematics, students practice skills in a variety of ways. One style of practice that can help students stay motivated and engaged is purposeful practice. Instead of a page full of repetitive problems, students focus on an activity that has a mathematical goal to achieve. Dan Meyer wrote a blog post a few years ago titled "Purposeful Practice & Dandy Candies" that started me thinking about how to make activities in my classroom more purposeful. Open MiddleOne of my favourite sources of problems with purposeful practice is Open Middle. There is a large selection of questions organized by topic and grade level. Each question has an "open middle" meaning there are many ways to explore and solve the problem. Below is a question submitted to the Open Middle site by Robert Kaplinsky. In this question, students try to find the arrangement of digits that yields the product closest to 50. Students will try numerous different arrangements of digits and get lots of practice multiplying decimal numbers without it seeming tedious. The question can also be quickly modified to give additional practice. For example, just add a hundredths place onto one of the factors and use 5 different digits. Math GamesGames can be a great way to encourage students to practice. There are lots of examples of but I'm going to mention just two. The first goes by several names. Joe Schwartz wrote a great post about Factor Captor. A similar game is described on the NCTM Illuminations site called the Factor Game. Students alternate turns playing on board filled with numbers. The first player selects a number to cover and adds that number to their score. The second player finds all the factors of that number, covers them and totals those number to add to their score. The roles are then reversed and play continues until there are no uncovered numbers remaining. There is a lot of math in this game and it is fun to play. The second game is Horseshoes from Math4Love. This game is played with a deck of cards numbered 1-9. Two cards are drawn to form a two-digit target number. Then four more cards are drawn. Players use these four digits to create an equation using addition and subtraction that is as close as possible to the target number. For example, let's say that the target number is 25 and the four digits given are 1, 3, 6 and 9. A student might create the equation 39-16 = 23. Another student might make the equation 13+6+9 = 28. There are lots of way to tweak this game for different levels of complexity. Both of these games allow for lots of numerical practice in a format that engages students. There are many excellent sources of ideas for mathematical games. My current 'go-to' resource is a book titled Well Played, 6-8. Math PuzzlesThere are several mathematical puzzles that include lots of practice with numerical computations in pursuit of a solution. KenKen puzzles and Maze 100 from NRICH are two such puzzles that I've used. I also think that Yohaku puzzles are great. They are numerical puzzles where you need to determine the number that is in each square in order to make the column and row sum/products.
Desmos ActivitiesFor classrooms with the available technology, activities from Desmos.com are another way to practice with purpose. An activity that incorporates a lot of meaningful practice is Marbleslides. In this activity, student try to capture stars on a Cartesian grid by creating a path using functions that marbles roll down. Students work to refine their functions to capture as many marbles as possible. Another activity that generates lots of practice is Transformation Golf. Students use a series of rigid transformations to move a shape to specified location. They have to find an efficient path around several obstacles. Each successive challenge increases in complexity. IncentivesThere are also activities that have a non-math goal for students to achieve. Lots of online math games have incentives for students such as badges to earn, experience points to accumulate or virtual prizes to win. There are also worksheets and activities with non-math goals. Worksheets such as "Algebra with Pizzazz" and "Punchline Algebra" have a riddle to be solved once all of the questions are completed. In my classroom practice, I used a number of activities with these types of incentives and I think that many students find them exciting and fun. If students are excited to do math, I consider that a win. However, I think that these types of activities should be used with caution. We don't want to inadvertently send a message to our students that math isn't fun by itself so we have to disguise it (like sneaking vegetables into their favourite foods so kids will eat them). Please Share!If you have a favourite math activity, game or puzzle that gets students practicing math with a purpose, I'd love to hear about it. Please leave me a comment. EL
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