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
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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.
Desmos Collections are a way of organizing and sharing lists of Desmos classroom activities aligned with curriculum outcomes. Below are links to my collections aligned with Nova Scotia mathematics courses. These are a work in progress.
Just going to leave this link here (from a tweet by Laura Wagenman) until I have time to turn it into an elementary Desmos collection: https://docs.google.com/spreadsheets/d/1wDbrLz5H3XmDLtpwDkkMdS3z5618SYMS8QdeOBUTAbw/edit#gid=0 Let me know if I've missed your favourite and I'll add it to my list. EL
Students work with scale in a number of mathematics courses in Nova Scotia. In the Mathematics Essentials 12 course, they work with scale in conjunction with reading blueprints. This scale activity uses the blueprints for an iconic Canadian airplane, the DHC3 Otter. I started the lesson with some estimation. I didn't let students know the name of the airplane to start with so that they don't just Google the name of the airplane and find out the actual length. Instead they use clues from the photo to make an estimate. Some references might include the size of the dock or ramp, the height of the wing, or the size of the windows or door. We use a routine I learned from Estimation180 to ask for an estimate they know is too low, an estimate they know is too high and then a just right estimate. After everyone had made and estimate, I showed them an image from a blueprint and asked them if they'd like to revise their estimate using some additional information. On the blueprint, they can see a scale drawn above the plane. Students use this information to either confirm or adjust their estimate. We can then use the scale to measure the image to find out the actual length of the plane (about 41 ft). After this opening activity, I showed them a picture of a 1/48 scale plastic model and ask them to tell me what the size of the finished model is knowing the size of the actual plane and the scale factor. The opening estimation allowed students a chance to think about scale. Next I handed out the blueprints for the plane. Our task was to use the blue prints to help create a 1/10th scale drawing of the top view of the plane using painters tape on the floor.
Students worked in small groups to recorded their measurements and calculate the measurements for the scale drawing. Once they were finished each group was provided with some painters tape and measuring tapes in order to make their scale drawing on the floor. It could be nice to have one group do their drawing on the wall, then it could stay up as a reference to their work with scale. We also talked about writing their measurements on the painters tape as they put it down. Variations and ExtensionsStudents who finish quickly can continue to add additional details such as the pontoons. This activity could also be done with larger groups creating a life size drawing of this plan using sidewalk chalk outside (weather permitting). NS Outcomes:
Mathematics Essentials 12  3.1 Calculate the dimensions of actual objects using blueprints with various scales Mathematics 11  M02 Solve problems that involve scale diagrams, using proportional reasoning Mathematic at Work 11  G02 Students will be expected to solve problems that involve scale; and G04 Students will be expected to draw and describe exploded views, component parts and scale diagrams of simple 3‑D objects. Mathematics Essentials 11  E4 create 2D scale diagrams and 3D scale models Mathematics 9  G03 Students will be expected to draw and interpret scale diagrams of 2D shapes. Summer is here and I've recently returned from a family vacation that included several lengthy flights. My son just turned eight years old and enjoys flying but can get a bit restless after a few hours on an airplane. Below are a few of the games and activities that we packed to make the time pass enjoyably. These games are best when they are small, lightweight, easy to pack and can be played on the limited space of an airplane tray table (often around 16.5" x 10.5" but there is no standard size). Games are ideally two player but if you're on a larger plane, you might be able to play a three player game. Pencil and Paper GamesGames that can be played with just pencil and paper are ideal for an airplane. We bring a few pencils and a tablet of paper with us for games, sketching or making notes. Here are a few of our favourite pencil and paper games:
Games with Dice, Cards and Counters
Commercial Board Games
Other StuffWhen on a long play ride I also bring snacks, a small LEGO set, and some art supplies to draw or colour with. Do you have suggestions for travel games to play with with your children? I'd love to hear your suggestions. EL
Some colleagues recently told me about an activity they had used in class called "Math Market". I'm not sure who originally created it. The teacher who shared it with me learned it at a math conference several years ago. I decided to give it a try with a Calculus class that was just finishing up a unit on integration. Here is how the activity is run. Students work in small groups (we had groups of three). Each group starts with $5 and selects a captain who can buy questions of different levels of difficulty from the market. Easier questions cost less and have a smaller profit. More difficult questions cost more and have a higher profit. The captain takes the purchased question back to their group to solve. Once they all agree on a correct solution, the captain returns to the market to sell the solution for a profit. The card is added back to the bottom of the market pile and some other group will have an opportunity to buy it. If their solution is correct, they buy a new question and continue working. If the solution is incorrect, they have to buy the question again to attempt a revised solution (or they can purchase a new question at a different level of difficulty). We decided to purchase the solution at a reduced price ($1 less) if they forgot to include the "+C" at the end for the constant of integration. The easiest questions were free so that if groups went bankrupt with an incorrect solution, they would still be able to "buy" another a problem. I printed the questions on coloured card stock and cut them out. Each question was marked with its level of difficulty. I also added a letter to the card so that it would be easy to find its solution to check the answers. Resources
How it WentI like that students got immediate feedback on their work. If it was wrong, they had to work with their group to correct their mistake. This was a test review for the class so there were lots of different types of problems mixed together and students had to determine what strategy would be best to solve each problem. It is a nice way to introduce some interleaved practice. This activity could be done with nearly any topic but it worked really well for integration as the questions were challenging and took them some time to solve. This made the market area less crowded.
I'm sure there are lots of variations of this activity. If you have some suggestions, I'd love to hear about them. EL
I recently attended Nat Banting's presentation at the Ontario Association for Mathematics Education annual conference in Ottawa (#OAME2019). He talked about "Creating Mathematical Possibility by Constraining Mathematical Possibility" (you can check out the slides from this presentation on his website). Last week I saw several educators sharing variations of a task that Nat presented on his blog called a 'menu task'. In a menu task, students are given a list of specifications and are asked to create functions that satisfy these specifications. It would be a fairly straightforward task for students to create a different function for each single specification (e.g. create a function that has a positive yintercept). Students are challenged by asking them to use as few functions as possible to satisfy all the specifications in a list (in whatever combinations they desire). Amie Albrecht shared a Linear Relationships version of this menu task which inspired me to try creating one for high school calculus. Calculus Functions Menu TaskInstead of jumping right into the menu, I wanted to make sure that students were familiar with the expectation so I decided to build up to the menu. I started by asking students to come up with one function to satisfy each specification. Then I used the same specifications but asked students to satisfy them with only two functions. After getting used to the idea, I then continued with the Calculus functions menu task. The students have just started integration so this is mainly a review with just a bit of integration thrown in. If you'd like to give this task a try with your students, here is a link to my google slides. Example SolutionsHere are three example functions that I came up with to satisfy these ten specifications (some specifications are satisfied by more than one function): A,D,E,H) y = (x+2)(x1)(x3) C,G,H) y = (x+2)(x)(x2) A,B, F, I,J) y = [(x+1)(x+1)(x1)]/(x1) Students were able to come up with fewer functions for this task. Here is one student's two functions to satisfy the ten specifications. EL
Over the recent holiday weekend, I brought out my copy of Dice Games Properly Explained by Reiner Knizia (@ReinerKnizia). I enjoy playing dice games at home with my family as well as playing dice games in class with students. Dice games often have simple rules and typically don't require of lot of material other than dice. Dice are a great mathematics manipulative that can be used with wide range of ages. Even very young students can practice subitizing numbers from 1 to 6 by reading the dots on the face of the dice. Below are a few of my favourite mathy dice games. Shut the Box
The object of the game is to cover as many of the 9 boxes as possible. Any numbers left uncovered at the end of your turn are added together. "Shutting the box", or covering all the numbers, leads to a perfect score of 0. The player with the lowest score is the winner. On the players turn, they roll the dice and add them together. You can then cover any boxes that are not already shut that sum to the total you rolled. For example, if I rolled a 2 and 4 my total is 6. I could cover the 6 box, the 5 and 1, the 4 and 2 or the 1, 2 and 3 boxes if they are uncovered. If I can't partition my number to cover any boxes then my turn is over and I add up any uncovered boxes to determine my score. This is a fun game with some strategy to it. It really focuses on early addition skills as well as partitioning numbers in a variety of way. It is a great game for a lower elementary classroom. NS Outcomes: Primary  N04 Students will be expected to represent and describe numbers 2 to 10 in two parts, concretely and pictorially. Mathematics 1  N04 Students will be expected to represent and partition numbers to 20. 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. Even Minus OddIn this game, players take turns throwing six dice. You then total all the even dice together and all the odd dice together. Subtract the odd total from the even total to get your score. Take counters from the centre of the table equal to your score. If you have a negative total, pay that number of counters to the centre (don't worry, if you don't have any counters left, you're still in the game). When all the counters are gone from the centre, the game is over and the player with the most counters wins. Start with about 10 counters in the centre or more if you have a large group playing. NS Outcomes: Mathematics 2  N02 Students will be expected to demonstrate if a number (up to 100) is even or odd. 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. Game of SixIn this category game, you need only one die and a score sheet. Players take turns rolling the die over six rounds. One your turn, roll the die and decide which category to score. Multiply the number on your die by the category value (1, 2, 3, 4, 5 or 6). Each category can be used only once each game. For example, if you roll a six on your first turn, you could score it in category 6 and earn 6x6 = 36 points. At the end of the six rounds, each player adds up their total points. The player with the most points wins. NS Outcomes: Mathematics 4  N05 Students will be expected to describe and apply mental mathematics strategies, to recall basic multiplication facts to 9 × 9, and to determine related division facts. Mathematics 5  N03 Students will be expected to describe and apply mental mathematics strategies and number properties to recall, with fluency, answers for basic multiplication facts to 81 and related division facts. NinetyNineThe focus of this game is on the order of operations for whole numbers. Five dice are rolled in this game and the player who rolls the dice calls out any number they wish between 33 and 99. This is the target number for the round. Once the dice are rolled, players create an expression using all five numbers on the dice and any operations (+,  , x, ÷). The goal is to create an expression whose value is as close as possible to the target number without going over. Divisions must work out without a remainder. Players secretly write down their expression. Once everyone has an expression (or a reasonable amount of time has passed), players reveal their expression. The player closest to the target scores a zero. All other players score the difference between their expression's value and the values that was closest (to a maximum of 5). Play as many rounds as their are players so each person can have a round setting the target. NS Outcomes: Mathematics 5  N03 Students will be expected to describe and apply mental mathematics strategies and number properties to recall, with fluency, answers for basic multiplication facts to 81 and related division facts. Mathematics 6  N09 Students will be expected to explain and apply the order of operations, excluding exponents, with and without technology (limited to whole numbers). A Focus on Mathematical ContentThere are lots of really fun dice games in Reiner's book although some are more suited to a mathematics classroom than others. I also really like the chapter on the theory of dice and probability (chapter 3). I think the games above are not only fun, but closely related to mathematics outcomes. A recent post I read from Hilary Kreisberg (@Dr_Kreisberg) discussed a protocol to assess good classroom tasks. One dimension of her protocol was to assess the mathematical content of a task and ask yourself if the task aligns well with specific gradelevel standards. I think this is an important aspect to remember and not just play games that are fun, but ones that also offer meaningful mathematical practice. EL
I recently had the opportunity to work with a calculus class on curve sketching and how derivatives affect the shape of a graph. The classroom teacher and I brainstormed some ideas about how we might infuse some hands on activity into the lesson. We decided to try an activity called Functions on the Floor. I originally saw an outline of this activity in a presentation from Liana Dawson called HandsOn Calculus Activities. Warm up  Functions and Their DerivativesWe started the class out with a Demos activity called Functions and Their Derivatives. We had students work in pairs on this activity. In the first part of the activity, students are presented with the graphs of three functions and they have to decide which is the original function, which is the derivative and which is the second derivative. In the second part of the activity, students create their own challenge and then get an opportunity to try out the challenges created by other groups. I really like the collaboration and discussions creating by working on these challenges. Functions on the Floor InstructionsAfter the warm up, we transitioned to the Functions on the Floor activity. We used masking tape to create several coordinate systems on the floor with the x and yaxis labeled from 3 to 3. At each of these stations was a list of information about a continuous function. Students used a small rope to create a function on the axis that satisfied all of the conditions listed. They then drew their function into a Desmos activity I had prepared. Using the drawings in the Desmos activity we could monitor students activity and plan for our review of the functions at the end of class. We heard some really constructive conversations taking place. The Google slides for the stations can be found here. Sharing our WorkI thought that this activity was a nice way to incorporate both handson physical problem solving while still leveraging some of the power of online tools like Desmos. I learned a few lessons doing this activity and the classroom teacher and I had a productive discussion after the lesson. We talked about how the lesson went and where we saw areas for improvement. We both agreed that eight stations was more than needed as it took some time for the discussions in the student groups to come up with a reasonable graph. We thought that some of the stations could have fewer constraints to consider as well. Also, I think next time I would use something besides rope. I had pretty inexpensive rope and it wasn't as pliable as I would have liked. I think a thick piece of yarn might have worked just a well. If you give this activity a try, let me know how it goes. Nova Scotia Mathematics Curriculum Outcomes Calculus 12 B15  Demonstrate an understanding of critical points and absolute extreme values of a function Calculus 12 B16  Find the intervals on which a function is increasing or decreasing Calculus 12 C5  Apply the First and Second Derivative Tests to determine the local extreme values of a function Calculus 12 C6  Determine the concavity of a function and locate the points of inflection by analyzing the second derivative EL

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