I had the great fortune to participate in the first Rosenthal Summer Institute in July 2022 at the Museum of Mathematics (MoMath) in New York City. I was one of 14 invited educators from across the United States and Canada. The institute was an opportunity to work with these educators to discuss innovative lesson design and to learn more about the Rosenthal Prize and the library of winning lessons from the past 10 years of the competition. Design PhilosopyDuring our time together, we had a number of discussions about designing lessons for students. One theme that emerged for me was the similarities between the design philosophy behind both the exhibits at the Museum of Mathematics and the Rosenthal Prize lessons. Below are a few of the shared characteristics that I noticed: 1. AnticipateWe had the opportunity to tour the MoMath exhibits with Tim Nissen, MoMath’s resident designer. Tim discussed the evolution of the design of several exhibits and how they had to anticipate how visitors, especially children, might interact with them. The safety of museum visitors is a critical factor. They had to anticipate how children might interact with the exhibits in unexpected or accidental ways to ensure that each exhibit was safe. Similarly, the Rosenthal lessons anticipated common misconceptions that students might have and how the teacher might respond to them in order to clarify or correct these misunderstandings. 2. DifferentiateMany of the museum's exhibits had descriptions of the mathematics principles involved displayed on a computer screen. The screens allowed the visitor to select one of three different levels of complexity for each description. This allowed each visitor to engage with the exhibits at the level to which they felt the most comfortable. Similarly, the Rosenthal lessons often contained directions for how the lesson could be adapted or extended for different students depending on their mathematical background. For example, in the Dice Auction lesson, there is a set of 15 "lots" to auction off. This is also an extended list of additional lots that can be used to supplement or adjust the number and complexity of the "lots." There are several additional "extensions and adaptations" listed for the activity that can be included as needed for the specific class you are teaching. 3. EngageA museum is all about public engagement and exhibits have to be entertaining and enticing. Exhibits should draw visitors in and captivate their attention. Exhibits also should be intuitive and easy to understand. An great example of this is MoMath's SquareWheeled Trike. Visitors can ride a tricycle with square wheels around a circular track made of inverted catenaries. Visitors easily understand the purpose of this exhibit. While they ride a host engages them in a conversation about the mathematics they are experiencing. The Rosenthal Lessons are also often designed to be engaging and interesting to students. They can draw students in and let them experience mathematics in a new or surprising way. The Random Walk lesson is a great example of this. Students can participate in a lesson that lets them physically see how random change can lead to surprising patterns. Rosenthal Prize LessonsIn order to make the Rosenthal Prize lessons easier to access for Nova Scotian/Canadian teachers, I have organized below where several of these lessons might best be placed within the Nova Scotia/WNCP curriculum. Mathematics 7
Mathematics 8
Mathematics 9
Mathematics 10
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Can't Stop is a board game that was designed by Sid Sackson and released in 1980. It is a "press your luck" dice game in which players roll for dice and arranges them into two pairs. The sums of these pairs allow the player to progress their markers up the columns labeled with the values from 2 to 12. The goal of the game is to be the first to reach to top of three of these 11 columns. The height of the column is related to the probability that the sum will be rolled. The more likely the roll, the longer the column is. I was excited to find a copy of this game at a second hand shop recently. I've played Can't Stop online numerous times (at Board Game Arena) but this is a game where playing inperson seems better than virtual. The physical process of rolling and arranging the dice as well as interacting with other players makes for a better experience. I think it plays best with 2 or 3 players. They game is made for 24 players but I've seen people use the square plastic pieces from the game Advance to Boardwalk to add additional players in different colors. The Great Races
A nice usercreated version of this paper and pencil game can be found at Board Game Geek. I like this version of the game because it is a quick game, with the "pushyourluck" aspect removed. Each player rolls just once, records their roll and then passes the dice to the next player. For a shorter game, you could play until 8 of the races are finished instead of completing all 11. Game at a GlanceBecause all the information about the game is contained on the game board, this is an excellent game for "Game at a Glance." Chad Williams has a collection of Game at a Glance images on his website, Beyond the Algorithm. Students are shown an image of a game in progress and could be asked a variety of questions.
I recently went camping and took this game along with me. I taught the game to several people (including kids) and because of the simple rules, they were all able to learn it in just minutes. Because all the pieces are plastic, they didn't blow away in the wind and if it got dirty, it could be easily rinsed off and cleaned. I think that this game would make a great addition to a mathematics classroom collection. What are your favourite games for the math classroom? Let me know! Nova Scotia Mathematics Curriculum Outcomes Mathematics 12 LR01  Analyze puzzles and games that involve numerical and logical reasoning, using problemsolving strategies. Mathematics at Work 12 P01  Students will be expected to analyze and interpret problems that involve probability. Mathematics at Work 11 N01  Students will be expected to analyze puzzles and games that involve numerical reasoning, using problemsolving strategies. 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. EL
In previous years, I've taught related rates in calculus class by the book. As in, we worked through the example problems from the textbook together and then students individually worked on the practice problems. After working through some examples, I would give students some more challenging problems as an assignment. Recently, a teacher reached out looking for some new ideas for teaching related rates. We brainstormed some ideas of how we could make related rates more engaging and handson. We decided to create some stations where students could experience and take measurements of related rates in action. It turns out, this is probably the reason that related rates problems are in our textbook. While doing some research to prepare this activity, I found an article titled, "The Lengthening Shadow: The Story of Related Rates" by Bill Austin, Don Barry and David Berman. This article is from Mathematics Magazine (Feb 2000, Vol. 73, No. 1, pp. 312). In this article, the authors state that related rates problems originated in the 19th century as part of a reform movement to make calculus more accessible. By observing changing rates, students would be able to measure concrete examples and discover their relationships. Joshua Bowman discusses how he grounded his teaching of related rates using observations in his blog post Using calculus to understand the world. Related Rates StationsFor this activity, we decided on four stations: Station 1  Blowing Up a Balloon. Blow 5 big breaths into a balloon. After each breath, measure the circumference of the balloon and calculate radius and volume. How are radius and volume related? Station 2  The Sliding Ladder. A metre stick is sliding down the wall. The bottom of the metre stick is moving away from the base of the wall at a constant rate. How fast is the top of the stick sliding down the wall? How are they related? Station 3  Building Fences. Build several “fences” (rectangles made with multilink cubes) such that the length is twice the width. Put them in order of size. Measure each rectangle’s length and width. What is the rate of change of the area? How are perimeter and area related? Station 4  Driving Cars. Two toy cars are traveling at different rates in perpendicular directions. How fast are each of the cars travelling? How fast is their distance apart changing? How are they related? After taking measurements at each of these stations, students drew a picture and created an equation to relate the quantities to each other. We used implicit differentiation to determine the relationship of the rates and then we tested our equations with our collected data. Reflection and ResourcesThe activity went a bit long for one class period. Next time I would either split the activity up over two class periods or reduce the number of stations to three. This will allow more time to consolidate the learning at the end of the lesson. As Tracy Zager says, "never skip the close." If you're interested in giving this activity a try, below are the files I used: Related Rates Stations Google Slides Related Rates Recording Sheet Update  13 AprilAfter doing this activity a few times in classrooms, I decided to reduce the number of stations from 4 to 3. In a 75 minute period, students were able to complete the three stations with a few minutes at the end to consolidate the lesson. I also changed the set up for the driving cars question to make it a bit more interesting. Here are the updated files that I've been using for 3 stations and a different problem for the driving cars problem. Related Rates Stations Google Slides Related Rates Recording Sheet Nova Scotia Mathematics Curriculum Outcomes Calculus 12 A3  Demonstrate an understanding of implicit differentiation and identify situations that require implicit differentiation Calculus 12 B14  Solve and interpret related rate problems EL
An "Add 'Em Up" activity is one where several problems are given and students sum the solution to come up with an answer. They can then check their answer to see if they have the correct total. If not, they can work to find their mistake. This "selfchecking" activity lets students know there is an error but not exactly where to find the mistake. There are lots of different versions of this activity. You can vary the number of questions that students have to add together. The more questions there are, the more difficult it will be to find any mistakes they might have. I feel that the harder the questions are, the fewer should be in a set(3 or 4 questions in a set seems to be the sweet spot). You could have students work in small groups or individually. You can give students several sets of questions all at once or sequence them one at a time or as a stations activity. Below is an activity I recently did with a Calculus 12 class to review several different derivative rules: Derivative Rules Add Em Up Activity Each of these sets of problems was printed on a different colored paper and each small group of students started with sheet 1. When they had a solution for the sum of the problems on their sheet, I checked their answer. If they were correct, I gave them the next set of problems. If they were wrong, I told the to find their mistake. If they were struggling I could give them a hint on where to look for their mistake or guide them with some probing questions about their work. The colored sheets made it really easy to scan the room to see how far each group had progressed and focus my attention on groups that may need some additional support. In the past, I've just printed the sum in the middle of the page but I really like having the check in with students when they have finished each set of problems. Students checked their final sum on a box with a 3 digit combination lock on it. If correct, they could retrieve a piece of candy for each member of their group from inside and then relock the box for the next group. For some other examples of "add 'em up" or "sum it up" types of activities, check out these links:
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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/notfunction 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
The holiday break is quickly approaching and playing games with your kids is a great way have fun as a family. These games are not only fun, but can also promote curiosity, strategic thinking and lead to mathematical explorations and discoveries. Here are a couple of my favourite sites where you can find well curated selections of games to play at home: There are lots of games out there, so how do you choose? Mark Hendrickson, in a recent Global Math Department webinar, talked about his philosophy about math games. He said that good games should be accessible to everyone, allow students to be curious, think strategically. While games can help reinforce mathematical skills, they must be fun primarily, not just a lesson disguised as a game. Here are a couple of my favourites that just need pencil and paper and are accessible by kids at just about any age:
Hope you all have a fun and festival holiday break. EL
Where in your city or region can you see the farthest? The higher up you are, either in a building or on top of a hill (or a building on a hill), the farther you will be able to see. If there are no obstructions, you can see all the way to the horizon. With the Pythagorean Theorem and the radius of the Earth (r = 6,371 km), a few calculations can reveal the distance to the horizon (d) for any height (h). Given the height of my eye above the floor, 174 cm, if I stand on the beach and look out over the ocean to the horizon, I should be able to see about 4.708 km. A recent podcast from "A Problem Squared" (episode 022) featured a similar question. The question that was submitted by a listener asked, "What's the furthest away you can see something from earth, that is also on the earth?" This question made me think about a new building being built near where I live in Halifax. Richmond Yards, at a height of 103.3 m, it will be the tallest in Atlantic Canada when it is completed. It is also built one of the highest parts of the Halifax Peninsula at 60 metres above sea level. I wonder if the view from the top of this new building will be the longest view in Halifax? Or could it even be the longest line of sight in the province of Nova Scotia? So if I stand at the top of the Richmond Yards tower (103.3 m), located on a hill that is 60 m above sea level, how far should I be able to see if there are no obstructions? Using the formula above, my new height would be 103.3 m + 60 m + 1.74 m = 165.04 m. Given that height, the distance to the horizon would be 45.858 km. That is quite an improvement. Do you think there is a spot in your town or region where you could see farther? Where in Nova Scotia can you look the furthest away at something else, that is also in Nova Scotia? Do you know where the highest point in Nova Scotia is? What about in furthest view in all of Canada? From the top of the CN Tower? From the top of Mount Logan, the tallest mountain in Canada with a summit of 5,959 m? What other factors effect how far you can see? What about seeing past the horizon to a tall building or mountain that sticks up over the other side of the horizon? NS Outcomes: Mathematics 8  M01 Students will be expected to develop and apply the Pythagorean theorem to solve problems. Mathematics 9  M01 Students will be expected to solve problems and justify the solution strategy, using the following circle properties: [...] A tangent to a circle is perpendicular to the radius at the point of tangency. EL
Selfchecking activities allow students to have immediate feedback on how they are doing. When these activities are completed in small groups, it gives students an opportunity for meaningful mathematics discussions. Students determine if they have the correct answers and if they don't, they can work together to determine where their mistake is. This allows the teacher to focus on groups that have misconceptions or misunderstandings as students will often find and correct their own computational errors. A selfchecking activity that I've recently been using is called "Odd One Out." I was inspired by a couple of activities that I found on TES. The first example was a page showing a number of expressions to evaluate using the order of operations. In the center of the page was a bank of possible solutions. There were 15 expressions and 16 solutions. The solution left over when all the expressions were evaluated was the "odd one out." The other example showed four sets of five linear equations to solve. All of the equations in each set had the same solution except for one. The goal was to identify the equation with the "odd one out" solution. Below are two "Odd One Out" problem sets that I created. The first one is to practice solving systems of equations and the second is to practice solving precentage problems. You could also use this activity as a quick warmup with fewer numbers. Jo Morgan (@mathsjem) shared an activity from MathsPad on her Math Gems #74 that had a sets of 9 radicals (surds): 4 simplified, 4 unsimplified and 1 odd one out. You could also use the same format as above with fewer questions. A couple of short examples below. Craig Barton (@mrbartonmaths) highlighted Odd One Out activities in a "Maths Resource of the Week" in late 2016. He describes a few different varieties of this activity and also points to a variety of different examples of this resource. Have you used an odd one out activity with your students? How did you use it? EL
Logical reasoning outcomes in the Nova Scotia mathematics curriculum involve spatial, numerical and logical reasoning. These are part of the grades 10 to 12 NS mathematics curriculum as well as the WNCP (Western and Northern Canadian Protocol) mathematics curricula. These outcomes focus on using puzzles and games as a vehicle for learning reasoning skills. Students are asked to determine, explain, and verify strategies for solving a puzzle or playing a game. They are also asked to identify mistakes in a puzzle or errors in a solution strategy. The NS curriculum guide reminds teachers that, "it is not enough for students to only do the puzzle or play the game. They should be given a variety of opportunities to analyze the puzzles they solve and the games they play. The goal is to develop their problemsolving abilities using a variety of strategies and to be able to apply these skills to other contexts in mathematics." Games are a great opportunity to build selfconfidence and a positive attitude towards mathematics. Games are inherently "low threshold high ceiling tasks". Students start by playing games at a basic level and as they gain experience, they develop more robust strategies for playing and winning. Opportunities to discuss games with other students help them to develop communication, decision making and reasoning skills. Games and Puzzles in DesmosDuring most years, puzzles and games offer an opportunity to change up classroom routines. Social interactions during game play help to build a positive classroom culture. Unfortunately, public health restrictions resulting from the COVID19 pandemic, have altered many classroom routines. In order to limit close interactions and sharing of manipulatives, many teachers have turned to online puzzles and games. While there are some great online resources for puzzles and games, many of them don't allow the teacher to observe and interact with students while playing these games. The teacher dashboard in Desmos activities allows teachers to synchronously observe and interact with students. Pacing tools give teachers control over how an activity progresses. Activity screens also allow teachers to include questions asking students to reflect on strategy or to find errors in puzzles. The dashboard can also allow teachers to give written feedback to students on their progress. Below are some Desmos activities that would accomplish spatial, numerical and logical reasoning outcomes. The choice of which category to place these games is sometimes quite subjective. For example, I think of a Skyscraper puzzle as using both spatial and logical reasoning and could go in either (or both) category. Here is a link to a Desmos collection with the games and puzzles mentioned below. Spatial Games and Puzzles
Numerical Games and Puzzles
Logical Games and Puzzles
Do you have a favourite Desmos logic puzzle? Please let me know about it so I can add it to my shared collection. NS Outcomes: Mathematic at Work 10  G01 Students will be expected to analyze puzzles and games that involve spatial reasoning, using problemsolving strategies. Mathematic at Work 11  N01 Students will be expected to analyze puzzles and games that involve numerical reasoning, using problemsolving strategies. Mathematics 11  LR02 Analyze puzzles and games that involve spatial reasoning, using problemsolving strategies Mathematic at Work 12  N01 Students will be expected to analyze puzzles and games that involve logical reasoning, using problemsolving strategies. Mathematics 12  LR01 Analyze puzzles and games that involve numerical and logical reasoning, using problemsolving strategies. EL
"Design and build a model birdhouse from a single sheet of 8.5" x 11" sheet of paper." This open ended activity seems simple at first but will require careful planning and attention to detail for students to be successful. You might start off this activity by showing a photo of an actual birdhouse and asking students to brainstorm the features of a good birdhouse. A website like this one might be a good guide. Next you can talk about the expectations for their model birdhouse design:
Students should then be asked to create a design. The design should minimize wasted paper (i.e. use as much of the page as possible) and be easy to assemble (i.e. minimize the number of pieces you have to cut out and assemble). You can then show students an example of a finished design. ProcedureStep 1  Students should brainstorm some possible designs (at least two) on a piece of looseleaf Step 2  Ask students to pick their favourite idea and share it with the teacher Step 3  Once the teacher approves their design, students are given a piece of card stock. They can then lay out their design with a ruler Step 4  When finished, students will measure and record all dimensions for their model. Students then calculate the surface area and volume of their design Step 5  The final step is to cut out and assemble their birdhouse model! Here is a Google slides document that could be used to introduce the activity to students and make the expectations clear. Math at Work 10 Activity: One teacher modified this activity by giving students a selection of designs to choose from instead of designing their own (here are links to pdf template 1 and template 2). Students then did all of the measurements and computations and had to determine costs for shingles on the roof, siding for the walls and paint for the interior. Here is a handout similar to the one she used. Extensions: If you were to take your model and use it to build an actual birdhouse from wood, what would have to change? By what scale factor would you have to increase the size? How would building with 3/4" thick wood (instead of flat paper) change the size of the pieces needed? What supplies would you need and how much would it cost to build? NS Outcomes: Mathematics 9  G01 Students will be expected to determine the surface area of composite 3D objects to solve problems Mathematics 10  M03 Students will be expected to solve problems, using SI and imperial units, that involve the surface area and volume of 3D objects, including right cones, right cylinders, right prisms, right pyramids, and spheres. Mathematic at Work 10  M04 Students will be expected to solve problems that involve SI and imperial area measurements of regular, composite, and irregular 2D shapes and 3D objects, including decimal and fractional measurements, and verify the solutions. Mathematics Essentials 12  2.4 Sketch and construct a model which will enable a student to show others some mathematics involved in a career interest EL

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