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
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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
How much does it cost to mail a package in Canada? What factors determine the cost of postage? If you have ever put a box in the mail at a Canada Post location, you'll notice the mail person do several things. First they put the box on a scale, then they measure the dimensions of the box with a measuring tape. Finally, they read the postal code for the destination and input it all into their computer. Then they'll tell you the price to mail it. Getting students to measure items with a variety of measurement tools is great practice to get familiar with metric and imperial measurement systems as well as becoming familiar with common measurement units. Sometimes this practice is just measuring lines on a worksheet or measuring objects around the classroom (paperclips, pencils, desks, etc.). This can seem a bit trivial. I was hunting for measurement with a purpose and thought of the measurements that postal employees do. It turns out, you can find the rate for mailing a package by using a page on Canada Post's website.
The students seemed to enjoy the activity and got some purposeful practice using metric and imperial measurements. Here is a link to the google slides for the activity and a recording sheet. NS Outcomes: Mathematics Essentials 12  1.6 Identify, use, and convert among and between SI units and Imperial units to measure and solve measurement problems Mathematics 10  M01 Students will be expected to solve problems that involve linear measurement, using SI and imperial units of measure, estimation strategies, and measurement strategies. Mathematic at Work 10  M03 Students will be expected to solve and verify problems that involve SI and imperial linear measurements, including decimal and fractional measurements. Mathematics Essentials 10  D3 estimate distances in metric units and in imperial units by applying personal referents. EL
Estimysteries are a math routine created by Steve Wyborny (@SteveWyborney). He has shared lots of examples of this routine on his website. An estimystery uses an image to invite students to make an estimate. After an initial estimate, clues appear which help students narrow down the possible set of answers and refine their estimates. Most of the estimysteries that Steve has shared ask students to estimate a collection of discrete objects. Metric MeasurementIn Nova Scotia, students need to be familiar with both metric and imperial measurements. In Math at Work 10, students need to be familiar with measurements as well as converting between units. While brainstorming with a teacher, we decided that an estimystery might be an engaging way to get students to think about different units of measurement while at the same time giving them some benchmarks for a variety of different units. Above are two estimysteries I created using Google slides. The first asks students to estimate the length of a pencil and the second asks students to estimate the weight of a bunch of bananas. Since I'm using these with high school students, the types of clues that are given can include vocabulary terms that you might not use with younger students. It allows a place to include a review of some terms that they may not have heard in a while (e.g. prime and composite). Updated Sept 9th  I created one more estimystery. This one to estimate the length of a bank of lockers in the hallway to the nearest hundredth of a metre. Updated Sept 16th  I created a fall estimystery. This one to estimate the weight of a pumpkin. Andrew Stadel's Estimation180 website is a great source of inspiration for these types of measurement estimysteries. For example, I used the image from Day 55, the capacity of a cylindrical vase, to quickly create an estimystery. NS Outcomes:
Mathematics Essentials 12  1.6 Identify, use, and convert among and between SI units and Imperial units to measure and solve measurement problems Mathematics 10  M01 Students will be expected to solve problems that involve linear measurement, using SI and imperial units of measure, estimation strategies, and measurement strategies. Mathematic at Work 10  M01 demonstrate an understanding of the International System of Units (SI) by describing the relationships of the units for length, area, volume, capacity, mass, and temperature and applying strategies to convert SI units to imperial units. M02 demonstrate an understanding of the imperial system by describing the relationships of the units for length, area, volume, capacity, mass and temperature, comparing the American and British imperial units for capacity and applying strategies to convert imperial units to SI units. Mathematics Essentials 10  D3 estimate distances in metric units and in imperial units by applying personal referents. D4 estimate capacities in metric units by applying personal referents. Stomp rockets are a fun project to explore STEM design at home. About $15 will buy all the materials you'll need to create this project (a few feet of plastic pipe, some pipe fittings and a bit of card stock will get you started). Materials
Rocket DesignThere are lots of things to consider when you design your rocket. How long with it be? How many and what size/shape of fins will it have to provide stability? Why does it even need fins (while most model rockets have fins, modern space launch rockets don't). What type of nose cone will it have (flat, round, cone, etc)? Will you add a small weight to the front of the rocket (such as a ball of cotton or bit of clay) to move the centre of mass forward? My favourite part of this project is the iterative design process. You experiment with all sorts of different design features and building techniques and then immediately test them out. My son and I have been building and testing rockets all summer. We've been trying out different rocket lengths as well as fin shapes and size. InspirationYou can be inspired by real life rockets or other objects you might see. The fins on the rocket below were inspired by the plastic fletching on arrows my son and I saw at a local store. Would arrow fins be a better source of inspiration than model rocket fins for paper stomp rockets? Where's the Math?
Over the summer, my son and I made a number of different rockets and tested them out. We were trying to launch are rockets as far as possible. We even bought a 100 ft open reel tape measure to help us measure distances. Our current house record is 158 ft (with a 2L bottle as the launch "engine"). Helpful LinksRocket Aerodynamics  https://www.sciencelearn.org.nz/resources/392rocketaerodynamics Rocket and Launcher Instructions  https://www.instructables.com/id/PaperStompRocketsEasyandFun/ Designing Your Own Model Rocket by D. Mark Ponder  https://ohio4h.org/sites/ohio4h/files/imce/Designing%20Your%20Own%20Model%20Rocket.pdf (this document has some great information about different nose cone and fin designs) NS Outcomes: Mathematics Essentials 12  1.6 Identify, use, and convert among and between SI units and Imperial units to measure and solve measurement problems Mathematics 10  M01 Students will be expected to solve problems that involve linear measurement, using SI and imperial units of measure, estimation strategies, and measurement strategies. Mathematic at Work 10  M03 Students will be expected to solve and verify problems that involve SI and imperial linear measurements, including decimal and fractional measurements. Mathematics Essentials 10  D3 estimate distances in metric units and in imperial units by applying personal referents. EL
Working from home and looking after an 8 year old at the same time throughout the past three months has proven to be complicated. Providing an enriching, engaging and educational home environment for my son while still attending to work deadlines and schedules has been a difficult balancing act. A provincial state of emergency required us to adapt our daily routines. One positive change has been playing board games. My son and I have played daily board games and card games. We have gone from simple children's games such as Snakes and Ladders to more complex Europeanstyle board games like Alhambra. Our favourites (and most played) games have been: Splendor, Tiny Towns, and Kingdomino. These games give players lots of opportunities to use spatial and numeral reasoning to make informed choices. These games all work well as two player games or with larger groups. Playing KingdominoA game that I have come to appreciate is Kingdomino. It is easy to learn, quick to play and has complex strategy. In this game, players take turns drafting domino shaped tiles to add to their kingdom. Players have to build within a size constraint. Players start with a 1x1 home castle tile and your kingdom, including this home tile, can not exceed 5x5 (meaning you can add a maximum of 12 tiles to your kingdom). This size constraint means that if your kingdom is not well planned, you might end up with empty spots that can't be filled. The goal is the maximize the value of your kingdom at the end of the game. To calculate the final value, you multiply the area of each contiguous region of a single terrain type by the number of crowns contained in the region (there are six different terrains types: pastures, wheat fields, lakes, mines, forests, and swamps). If a region doesn't have any crowns in it, it scores zero points (e.g. there is a region of 4 lakes in the kingdom below but because it has zero crowns, it scores 4x0=0 points). There are four different regions with crowns in the kingdom below. The wheat field (outlined in orange) scores 4x1=4 points. The pasture (outlined in blue) scores 6x6=36 points. The forest (outlined in red) scores 5x2=10 points. And finally, the mine (outlined in green) scores 1x1=1 point. This gives a total values of 4+36+10+1=51 points... a decent total score. Bringing Math to the SurfaceWhen I first looked at the mathematics applied in this game, I only considered adding up the scores to determine a winner at the end. Playing this game with my son however led me to see how mathematical decision are used throughout the game. Each turn, you must decide which domino (or dominoes if you're playing with two people) you will select for your kingdom. In order to bring mathematical reasoning to the surface, I began reasoning out loud. This let my son hear all the considerations that go into my selection. Given the growing kingdom below, which of the four dominoes would you select and why? The first domino doesn't have any crowns, but it could be added to the bottom of the kingdom to make a region of 4 wheat fields where crowns could be added on a future turn. The first domino also gives me the benefit of going first next turn (dominoes are placed in order from top to bottom each turn). The second domino has a crown in a forest. This could be added to existing region of 3 forest that has 2 crowns already. This would turn your forest region from scoring 3x2=6 to 4x3=12... a gain of 6 points. The third domino would be even better. Playing it in the bottom right would still gain you the 6 points in the forest region but also turn your 2 point lake region in to a 3 point lake region. The last domino could add a crown to your lake region and a space to your forest region. This would make the forest region worth 8 (a gain of 2) and your lake region worth 6 (a gain of 4). Looks like the third tile would give us the most immediate gain in score (a gain of 7). ProbabilityOlder players will also need to consider the unequal distribution of terrain types and crowns. There are lots of wheat fields (26) but only 5 have a single crown. There are 14 pastures but two of these have one crown and two have double crowns. Mines are rare with only 6 of them in the game but they can generate big scores. Three of the mines have double crowns and one mine has the only triple crown in the game. The trick is to grab them when you can and place them next to each other. Strategy vs. LuckThere is a fair amount of luck in this game. The dominoes are shuffled and show up in random order. If you have the first pick in the round you can get a great tile but if you go last, you may get stuck with a tile that you don't really want. Also, if you're not playing with four players, you won't be playing with the full set of dominoes. A domino you were hoping to get might not even be in the game. With that being said, there is enough strategy to keep both kids and adults thinking critically about their decisions throughout the game. Playful MathI think this is a great example of a board game that demonstrates a playful and entertaining way to practice mathematical skills and reasoning. It's easy to learn rules and quick play time make it a great choice for playing with younger family members. It has been in regular rotation with my family during our at home learning. If you want to quickly learn how to play Kingdomino, you can watch Rodney Smith's Watch It Played video on YouTube. I have come to appreciate how much skill there is to teach an 8 year old, with limited patience, the rules to a new board game. If you want some tips on teaching your kids (or friends) how to play a new game, I highly suggest Shut Up & Sit Down's video How to Teach Board Games Like a Pro. A lot of these tips apply equally to the classroom and teachers will be able to see how good pedagogy is universal. EL
During this time of increased social distancing, there are some things that I'm doing a lot less of and a lot less often. For example, I rarely drive my car despite the low price of gasoline. Other things I'm doing a lot more of. I've played more board games in the last three weeks than I have in the last year. Another thing I've been doing more of is interacting with people virtually through video meetings and social media. I have really enjoyed taking part, along with my son, in Annie Perkins (@anniek_p) daily Math Art Challenge on Twitter (#mathartchallenge). You can check out the entire list of math art challenges at https://arbitrarilyclose.com/home/. I've also been reading electronic newsletters from Kent Haines (@KentHaines) and Dan Finkel (@MathforLove) on mathematical games and activities for all ages. I recently started reading the weekly MindBenders for the Quarantined! from MoMath's puzzle master, Dr. Peter Winkler. There are many talented people creating opportunities and resources for students and teachers to engage with mathematics in new ways. A recent Math Art Challenge from Annie Perkins made me think about how curiosity can sometimes lead to some amazing connections. It also highlighted for me how having a breadth of experiences and depth of mathematical knowledge can help me pause and notice these connections. I did this challenge with my son. He got frustrated with this challenge in that his circles were not exact (he is 8 years old). I started to show him how to use a compass to make the circles. We worked on it for a while together. He moved on to another project and I finished by adding a bit of color. Later, I was reading the Games for Young Minds newsletter in which Kent Haines suggested Sierpinski triangles as a math project for kids that helps reveal the beauty of mathematics. I noticed the connection here between Annie's art project with Apollonian Gaskets and Kent's suggestion of Sierpinski Triangles and then started thinking about all of the other ways that this simple art project might connect to a host of different mathematical ideas. Here is a short list of some of the related connections that I saw or explored after doing this project:
I notice that many of these topics are not explicitly in the mathematics curriculum. That being said many of them can be a way to look at curriculum outcomes from a different perspective or as a way to practice fundamental skills. What other things come to your mind when exploring this activity? When have you discovered unexpected mathematical connections when exploring a new activity? EL
I recently had the opportunity to try out a digital breakout with a Precalculus 12 math class. The classroom teacher and I wanted to create an opportunity for students to have some interleaved practice as a cumulative review for the course. We liked the idea of a breakout game but we wanted to make sure that all of the students got a chance to do a wide variety of problems. Our solution was to do a digital breakout in small groups of 23 students. This was the first time that I had created a digital breakout game so I went hunting online for some examples that might spark some ideas. I found Tom Mullaney's (@TomEMullaney) Digital Breakout template page to be very helpful in figuring out what I was going to do. It gave me lots of ideas and inspiration. I also found I found a post from Meagan Kelly (@meagan_e_kelly) showing an example of a math digital breakout that I was just what I was looking to do. I learned how to create a google site and conquered a number of new technical challenges. While creating the site took some effort, the classroom setup was easy and there were no materials required. I thought the breakout went well. The students were very engaged and they reviewed lots of different concepts from throughout the year. They liked working in groups and having a variety of different types of puzzles to solve. Many students were consulting their notes and examples from the textbook to find solution strategies. They were also using online tools like https://www.desmos.com/ to help them graph and visualize mathematical relationships. All the problems were selfchecking. If the combination for a lock didn't work, they knew that they had made a mistake and had to work together to find and solve it. They also all got to work at their own pace. To add a bit of additional flair, we added a final physical lock and box for students to unlock with a small treat inside. If you'd like to give this breakout a try, check it out. The link is: bit.ly/PC12Breakout. EL
I recently read the book Taking Shape: Activities to Develop Geometric and Spatial Thinking (2016). This book is focused on developing spatial reasoning for P2 students and was written by Joan Moss, Cathy Bruce, Beverly Caswell, Tara Flynn and Zachary Hawes. The book comes as the result of the ongoing Ontariobased Math for Young Children (M4YC) research program. I worked through some of the ideas and activities from Taking Shape with elementary educators and I would highly recommend it. The activities in the book are grounded in research and honed in elementary classrooms. I especially like the emphasis on playful pedagogy, "a method of playing with mathematical ideas that is intriguing, engaging, challenging, and joyful." A number of studies have recently been published which focus on the benefits of spatial reasoning for elementary students and how spatial reasoning and mathematical thinking are linked. Mark Chubb (@MarkChubb3) wrote about onehole punch puzzles on his blog based one such study (Tom Lowrie, Danielle Harris, Tracy Logan & Mary Hegarty (2019): The Impact of a Spatial Intervention Program on Students’ Spatial Reasoning and Mathematics Performance, The Journal of Experimental Education, DOI: 10.1080/00220973.2019.1684869). Check out his blog for a description of the study and activity. I was disappointed after reviewing this research is that it didn't discuss how they determined which spatial reasoning tasks were used in the study interventions. I was hoping that there would be a discussion of the characteristics of an effective spatial reasoning task. How do you evaluate spatial activities in order to determine which activities might give the most benefit? I was listening to the Numberphile podcast with Grant Sanderson (of 3blue1brown) recently and he stated, "being forced to articulate something clarifies thought." In that spirit, I'm going to try to articulate the characteristics I use when analyzing spatial tasks. Characteristics of an Effective Spatial Task
Here are a few games and tasks that satisfy my characteristics above:
Do you have a favourite spatial reasoning task or activity? Please share with a comment.

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