Below is a simple probability game that I've played with students for many years in a number of different courses. I don't really remember where I got it from but if you've seen it before, I'd love to know where it originated. I've found that students enjoy it and they really develop some good ideas about probability. Its similar to the Two-Dice Sum Game from Marilyn Burns but with some additional mental math for older students. ## InstructionsThis is a game for two or more players. Each player creates their own score card by drawing two concentric circles and then divide it into eight sectors. Choose eight different numbers between 2 and 12 and place them in the sectors in the outer ring. In the inner ring of the circle place numbers that add up to 100 (it is easiest to stick with multiples of 5). Roll two dice and add the numbers together, if the total is one of the numbers in the outer ring of your score card, you score that numbers value (the number in the inner ring). Players alternate rolling and scoring on their card until one of the players has 150 or more points (or to 100 for a bit shorter game). A google doc handout for this is available here. ## Sample Score CardTypically, the first time students play, they will randomly select numbers and values for each sector. After students have had the chance to play a few games, stop and ask them to look at different score cards to see if there any common characteristics of winning cards. What makes a good score card? Is it all just luck or are some cards better than others? ## Developing a StrategyWhat makes a good score card? Here are some common student observations: - Assign the highest point values to the totals that come up the most often.
- Don’t include 2 or 12 on your card at all since they come up so little.
- 7 should have the highest score value since it comes up more often than any other number.
## Experimental ProbabilityYou might ask students to roll two dice a bunch of times and record which numbers come up each time. Create a bit dot plot or a bar graph at the front of the classroom using all of this data. Ask students to predict what the shape of this graph will be. You could also use an online tool to model lots of dice rolls to see how it compares to the class data. You could then discuss the theoretical probability for each sum and compare it to the data you gathered. I always find this a fun activity and a nice way to start a discussion about probability and strategy. A nice question to start the class on the following day would be a Would You Rather? math prompt, "Would you rather flip 2 coins and win if they match OR roll 2 dice and win if they don't match?" Another great follow up activity would be Don Steward's Dice Bingo.Nova Scotia Mathematics Curriculum Outcomes Mathematics at Work 12 P01 - Students will be expected to analyze and interpret problems that involve probability. Mathematics Essentials 10 G4 - Compare predicted and experimental results for familiar situations involving chance, using technology to extend the number of experimental trials. Mathematics 8 SP02 - Students will be expected to solve problems involving the probability of independent events. Mathematics 7 SP06 - Students will be expected to conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or other graphic organizer) and experimental probability of two independent events. EL
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About a year ago, I signed up for Samira Mian's Udemy course on Islamic Geometry. I also purchased a copy of Eric Broug's book Islamic Geometric Patterns. I wanted to learn the basics so that I could determine if this might be a good way to satisfy the grade 7 mathematics geometric constructions outcome. I designed a short unit that I described last year. Recently, I decided to try replicating some of these patterns using the online Desmos calculator and geometry tool. I think having some experience drawing these patterns with a compass and straight edge was helpful. If you're looking for some Islamic geometric patterns to try, YouTube is a great place to get some ideas. There are some great instructional videos from Samira Mian and Nora Youssef, among others. The first pattern that I tried was a Star and Hexagon pattern that I learned from Samira's Udemy course. I learned that sticking with exact values are worth the effort. Rounding intersection points and slopes of lines to the nearest tenths or hundredths place work well at first but the errors compound and things start to get messy down the road. Interlacing the pattern gave me lots of practice with domain and range restrictions. ## 8 Fold RosetteNora Youssef has a nice video tutorial on for drawing an 8-Fold Rosette pattern. I did this pattern twice. The first time I constructed the basic pattern and the second time I added interlacing. I used the polygon function to add colour and figured out how to use trigonometry to rotate the polygons around the origin. This made it really efficient. I created a table with the vertices of the polygon and then just duplicated and rotated that polygon around the rosette. I duplicated the polygons multiple times to make the colours bold. Links: - 8 Fold Rosette - https://www.desmos.com/calculator/hs2hguhs29
- Interlaced 8 fold Rosette - https://www.desmos.com/calculator/fy6zoierjn
## Mathy MomentsYou can see from my notebook below that some of the math took me a few tries (this goes on for several pages). To make the weave for the 8 fold rosette, I made lines parallel to the original with a distance of 0.5 above and below. Each ribbon was then 1 unit wide. I was working with the equations in point-slope form. I'm pretty sure that there are more efficient ways to do these calculations but I haven't discovered them yet. I really like how these messy bits encourage me look for more efficient and elegant methods. ## Desmos Geometry ToolAfter working with the Desmos calculator for a while, I wanted to give the geometry tool a try. I decided to try a pattern that I saw on the Pattern In Islamic Art website. This site has some great resources. The pattern that I tried was from David Wade's book Pattern in Islamic Art. The geometry tool requires much less algebraic manipulation, but I find hiding the underlying grid is much more tedious than in the calculator. Everything has to be hidden individually instead of turning a whole folder on or off in the calculator. I've drawn this pattern in the past by hand and it would have been much more difficult if I didn't have that previous experience. ## Future ProjectsI've tried tiling some designs to cover the plane but I haven't come up with any good methods for this yet. I've also tried using sliders to dynamically adjust some of the relationships between the sizes of the pieces in these designs. These are great challenges and are helping me learn new features of Desmos. Dan Meyer wrote "If Math Is The Aspirin, Then How Do You Create The Headache?" I hesitate to call these graphing projects "headaches" because I enjoy the challenge. Regardless, this is a case where my need for mathematical solutions guide my learning and give me reasons to explore new graphing methods. EL
I recently had the pleasure of attending the Nation Council of Teachers of Mathematics (NCTM) Annual Meeting and Exhibition in Washington, DC. This was my first big math conference. I attended some great presentations. One message that I heard reiterated in a number of presentations and one that resonated with me was about rich tasks. The message was that having a rich task is only a starting point for effective mathematics instruction. A rich task in math class is like being dealt a great hand of cards in Poker. It makes winning easier but it still takes a seasoned player with solid understanding of the complexities of the game to win a big pot. ## Full Stack LessonsDan Meyer presented a session called "Why Good Activities Go Bad" in which he discussed a math task called Barbie Bungee. After giving summary of the task he interviewed three different teachers on their use of the task and their students' experiences. Dan asked us to think about what makes a task engaging and productive and what might make it fall flat. He talked about "full stack" lessons and how the mathematical task itself is just one component of a fully developed and presented lesson. ## Engaging StudentsSara VanDerWerf, in her presentation ‘Engaging Students in Seeing Structure’, talked about her overarching goals when lesson planning. Using routines such as Notice and Wonder and Stand and Talks, Sara supports her students to see and talk about math concepts before they are formalized. Students have a chance to engage in the mathematics and build conceptual understanding. These types of routines which allow students to be curious about math and ask questions are important elements for getting the most out of a rich math task. ## Super (Secret) Mathematics of Game ShowsOne of the most engaging sessions that I went to at NCTM was presented by Bowen Kerins. He had a fun presentation called Super (Secret) Mathematics of Game Shows. A few elements of his presentation that stand out: - Setting the Stage - Bowen had fun game show music playing as people entered the room to set the stage. A simple music cue, stopping the music, let us know that the presentation was beginning.
- Audience participation - Bowen invited "members of the studio audience" to participate in mock game shows to demonstrate how the games are played. It was also a bonus that he had prizes to give away that gave real stakes to playing these games... winning mattered.
- Engaging mathematics - the mock game shows contain really interesting and at times counter-intuitive mathematics. Learning about the math behind these games made me feel like I was getting privileged information.
## Rich ConversationsShortly after returning from NCTM, I saw these tweets above from Fawn Nguyen and from Cathy Marks Krpan. These thoughts sound like they reflect many of the messages that I heard in Washington, DC this year. This morning, I also saw John Rowe's blog post "The Secret Sauce of Great Lessons." It looks like he attended several of the same sessions that I did and had a similar reflection. ## My Lesson Planning ChallengeOne the first slide of my presentations, down in the notes, I often write, " The essential components of a presentation: a clear focal point, a strong flow and structure, a beautiful design and a compelling delivery." (I picked this quote up here). It serves as a reminder to stay focused and think about the structure of my presentation (or blog post for that matter). I need to create a similar reminder for planning lessons to focus on more than the task itself. I need to consider how that task will be implemented to make it as engaging and productive for students as possible. EL
I recently did an activity with students to answer a question by collecting and analysing data. I was inspired by similar activities from Bruno Reddy, Mean Paper Aeroplanes, and Julie Reulbach, Paper Airplanes for Measures of Central Tendencies. We started class by watching a video of the Paper Airplane World Championship - Red Bull Paper Wings 2015. This short video (about 3 minutes) shows the highlights of three paper airplane competitions; Distance, Airtime, and Aerobatics. After watching the video I let students know that we would be making paper airplanes for a distance competition. |

We started by brainstorming with students the characteristics of a good paper airplane that will fly a long distance. Most students have some experience in making paper airplanes. Several features that we discussed were: the shape (a glider or dart shape, wing angles), design features (symmetry, vertical flaps or a heavy nose), construction methods (sharp creases and accurate folds), materials (type of paper) and how it is thrown (launch angle, strength and accuracy of the thrower). |

## Predictions

## The Experiment

## The Results

## Reflection

Students had a handout where they were asked to reflect on how they could make this experiment more accurate and reliable. We thought that a few practice throws would help before we started collecting data (most student's last throw was their farthest distance). Another option would be to have more than three trials for each plane to increase the reliability of the data collected. There were lots of other really good suggestions as well. |

**Nova Scotia Mathematics Curriculum Outcomes**

**Extended Mathematics 11**

**S01**- Analyze, interpret, and draw conclusions from one-variable data using numerical and graphical summaries.

**Mathematics 9 SP03**- Students will be expected to develop and implement a project plan for the collection, display, and analysis of data by: formulating a question for investigation; choosing a data collection method that includes social considerations; selecting a population or a sample; collecting the data; displaying the collected data in an appropriate manner; drawing conclusions to answer the question.

## Math Movement Routines

**Stand and Talks from Sara VanDerWerf**- Sara describes a tweak to the standard “Think/Pair/Share” routine that has greatly improved the number of students participating in her classroom discussions. This strategy incorporates movement that gives every student a chance to talk out loud and share their ideas without distractions.

*“Learners, I’d like everyone to stand up. Do not have anything in your hands. No calculators. No notebooks. No phones or pencils. Nothing. In a moment I am going to give you something that I want you to look at with a partner... I want to hear you asking each other things you wonder about. Please go now and find your partner.”*

**Vertical Non-Permanent Surfaces (VNPS) from Peter Liljedahl**- Students work in small groups standing at vertical non-permanent surfaces such as chalk boards or dry erase boards around the perimeter of the classroom. This allows the teacher to easily see what each group is working on and encourages discussion. The non-permanent nature of the surface lowers the risk of making mistakes and prompts students to start working faster and persevere longer. Check out posts from Alex Overwijk, Nathan Kraft and Laura Wheeler discussing this instructional strategy.

**Math "Scavenger Hunts" / Circuit Training / Around the World**- The idea of a math scavenger hunt is that questions are posted on the walls around the room. The answer to each question will lead to the next question. Students move from question to question until they have completed the loop. The activity is self-checking because if students don't find their answer then they know they've made a mistake and need to work to correct it. I've blogged about this activity in the past and have used a number of variations of it. A teacher recently showed me a variation of called Math Stations Maze where the questions are numbered and the multiple choice answers lead to the next station similar to a choose your own adventure novel.

**Math Stations**- This instructional strategy does not have to be complicated. I've seen teachers successfully push some desks together to make stations and put some math problems at each station. Students work in small groups completing the questions at their station and move to the next station when finished. I like to have one more station than there are groups so that there is always an open station to move to. This helps to minimize wait time between stations. It takes a bit more up front work, but I like to have an add-em up activity at each station so that students immediately know if they've answered the questions correctly. With this type of activity, the sum of the answers to several problem are given. If the students sum is not the same, they know that there is a mistake and work together to find where it is.

**Take Your Class Outside**- Every once in a while, when the weather is nice, it is great to get outside. Stock up on some sidewalk chalk and hit the pavement to do some math. Not only do students get up and moving, they get some time to practice as well as decorate the playground or sidewalk with beautiful mathematics.

## Outcome Specific Examples from the NS Curriculum

**Grade 9 G02 Similar Polygons -**You might measure the height of a flagpole or other tall object using a mirror and similar triangles. Here is a tweet showing photos of students engaged in this activity.

**Grade 9 PR07 Linear Relations**- Barbie Bungee. Students determine a relationship between the number of rubber bands used in a bungee cord and how far a Barbie/action figure will fall. They this use this information to extrapolate how many rubber bands to use for an exciting bungee jump from a set height.

**Grade 8 PR01 Linear Graphing**- Body Graphing/Algebra Walk. Set up a large Cartesian plane on the ground (outside with chalk or indoor with painters tape). Ask students to choose a spot on the x-axis, (and make sure that some choose negative numbers). Take a portable whiteboard with you (or you could write out some functions on paper in advance), and write a linear function (for example y = 2x + 3). Ask each student walk to their correct (x, y) coordinate given their starting x value. Here is a video from the Teaching Channel demonstrating a similar idea. There is also a description and video of this activity from Martin Joyce (@martinsean).

Grade 8 M03 Surface Area - Cover a filing cabinet with sticky notes to determine its surface area. This is a great 3-Act Math task that you can tackle in your classroom. You just need a healthy supply of post it notes and you estimate and then cover just about any right rectangular prism, right triangular prism, or right cylinder in your classroom or school. |

**Grade 7 N01 Divisibility**- Divisibility Hop. Stations are set up around the room, each with a 3 or 4 digit number. The teacher calls out a number. If students are standing at a station with a number that is divisible by the number the teacher called, they hope to a new station.

**Grade 7 G03 Transformations**- Have students use the floor tiles as a coordinate plane then graphing various transformations with painters tape.

**Grade 7 N07 Fractions and Decimals**- Put(t)ing Rational Numbers in Order. Teams of students work to put rational numbers and decimals in the correct order and then putt a golf ball in the style of the Price is Right television show Hole in One game. You could alter this game to put the rational numbers on cards and deal cards to each person on a team and then they have to stand in the correct order.

**Grade 7 PR07 Equations**- Linear Equation Groups. Introduce this activity by calling out a small positive integer (for example 3). Students move about the room to form groups of that size. Once students are familiar with this activity, you can call out and/or write a linear equation on the board (for example 2x + 1 = 7). Students solve the equation and then move into new groups of that size.

## Step 1 - Math Convention Carry

- Baron Fig Vanguard softcover notebook with dot grid paper. Dot grid is perfect for me for both taking notes and working on math problems.
- Staedtler 15 cm ruler and 60 degree triangle
- Staedtler Single Hole Mini Metal Pencil Sharpener
- Koh-I-Noor Dry Marker Highlighter Pencils
- Tombow Mono 100 HB Pencil. This is my preferred pencil for taking notes due to its bold line and good point retention.
- General's Cedar Pointe Pencil #1/#2. I reach for this pencil when I have a tough math problem to solve or geometric figures to sketch. It writes smoothly, sharpens well and has an excellent eraser.
- Desmos Pencil... because it's a math conference. (provided by my colleague @Kelly_Zinck from a previous NCTM conference)
- Sakura SumoGrip Retractable Eraser. Sometimes life gets messy.
- Pentel Tradio EnerGel Pen - Black Ink. For filling out documents or signing autographs.

## Step 2 - Pick Out Too Many Sessions

## Step 3 - Fine Tune My List of Sessions

I find it really hard to narrow down the sessions that I'd like to attend. There are a lot of amazing presenters and really interesting and informative topics that I would like to learn more about. Unfortunately, I can only be in one place at a time so I have to make some hard choices. For each time slot, I'm selecting a first choice and a back-up session because that's the advice I've been given by more experienced conference attenders. |

## Step 4 - Listen to Good Advice

If you have any good advice from a first time NCTM conference attender, please let me know. I'm moving right along and perhaps I'll see you there!

## Sharing Candy

## Ripping Tickets

*n*, its partition number,

*p*(

*n*), is defined as the number of ways we can write it as a sum of positive integers. For example, since the number 3 can be written as three different unique sums (1+1+1, 1+2 or 3), we say that

*p*(3)=3. If we were looking for the total number of ways to partition the twelve tickets into any number of groups, our answer would be p(12) = 77 (

*from OEIS A000041*). In our problem above however, we're looking for the number of ways to partition 12 into exactly 4 positive integers. We can do this either by counting with an organized list (brute force) or using recursion. For a description of the recursion method, see https://brilliant.org/wiki/identical-objects-into-identical-bins/.

Organized Counting1) 1+1+1+9 2) 1+1+2+8 3) 1+1+3+7 4) 1+1+4+6 5) 1+1+5+5 6) 1+2+2+7 7) 1+2+3+6 8) 1+2+4+5 9) 1+3+3+5 10) 1+3+4+4 11) 2+2+2+6 12) 2+2+3+5 13) 2+2+4+4 14) 2+3+3+4 15) 3+3+3+3 | Recursive Methodp(12,4) = p(8,1) + p(8,2) + p(8,3) + p(8,4) p(8,1) = 1 p(8,2) = 4 ( when r = 2, p(n,2) = n/2)p(8,3) = p(5,1) + p(5,2) + p(5,3) p(8,4) = p(4) = 5 ( when n <=2r, p(n,r) = p(n-r))p(5,1) = 1 p(5,2) = 2 ( when r = 2, p(n,2) = n/2)p(5,3) = p(2) = 2 ( when n <=2r, p(n,r) = p(n-r))Therefore, p(12,4) = 1 + 4 +(1 + 2 + 2) + 5 = 15 |

## Exploring Problem Structures

*Same Surface, Different Deep*Structure math problems. These are a set of problems (typically four) that have a very similar context but different solution strategies. The intent is for students to focus on determining the structure of each question and then to identify the corresponding strategy needed to solve it. I think this is an interesting routine for mathematics outcomes where there are a large variety of similar structures (like solving quadratics word problems or combinatorics problems).

**Nova Scotia Mathematics Curriculum Outcomes**

**Mathematics 12**

**P05**- Solve problems that involve permutations.

**Mathematics 12**

**P06**- Solve problems that involve combinations.

Pentominoes are a great math manipulative that deserve a place in just about every math classroom. A pentomino is a shape created by connecting five square together. I think that they are underappreciated in their simplicity and adaptability. Asking small groups of students to work together to discover the twelve unique pentominoes is a rich math problem that invites thinking and discussion. |

1. Precious Pentominoes from Mike Jacobs (@msbjacobs) - Mike created this activity which asks students to take any two of the twelve pentominoes to create a closed shape with symmetry. Then students are asked to calculate their shape's value using the formula: Value = (Perimeter + Area) * Number of Sides. Students are asked to find the shape with the largest value possible. I also ask students to find the shape with the smallest possible value. There are some great opportunities for students to talk about symmetry, use mental math and think about strategy. |

**Pentomino Puzzles**from Jon Orr (@MrOrr_geek) - Jon describes an activity where students are given a hundreds chart and a transparent pentomino. Ask students to place their pentomino on the hundreds chart so that it covers a sum of 135. Then ask students to share their strategies for finding the correct placement. Are there sums that are not possible to achieve? Continue the activity by selecting different tiles, giving different sums to find, creating equations and solving them. I also saw these questions posted by Amie Albrecht (@nomad_penguin) on twitter, "Can you find a pentomino that covers numbers that: sum to 150, sum to an even number, covers three multiples of 4, and more". Jon created an online Desmos Activity that allows you to do this activity virtually.

**Pentomino Farms**- I was introduced the the pentomino farms activity from Martin Gardener's book

*Knotted Doughnuts and Other Mathematical Entertainments*. The task is to use the 12 pentominoes to build a fence around a field on your farm. The rule used to join the pentominoes to form a fence is that they must touch along the full edge of a square and not just at the corners. There are four types of farms to create. For each type, what's the largest field you can enclose?

- A fence of any shape enclosing a field of any shape.
- A fence of any shape enclosing a rectangular field.
- A fence with a rectangular border enclosing a field of any shape.
- A fence with a rectangular border enclosing a rectangular field.

One way to play this game to lay the pentominoes on a piece of chart paper with the same grid size as the pentominoes (typically 1 in. squares). This helps in counting the area of the field. Another suggestion is to use the pentominoes and board from a Blokus game. This keeps the pieces from sliding around as the fence is constructed. |

**How Convex is a Pentomino?**- Which pentomino shape is the most convex? How do you measure "convexity"? This is an interesting question that generated lots of discussion on Twitter. Some interesting methods of measuring this were discussed by Alexandre Muñiz (@two_star) in his blog post, "Vexed by Convexity." I think it would make a great discussion for students to rank the pentominoes by convexity and then defend their choices.

5. “Golomb’s Game” from Solomon Golomb - There are several variations of this game including commercial versions. The game is played with two or three players on an 8 x 8 grid. Players take turns placing pentominoes on the board so that they do not overlap with existing tiles and no tile is used more than once. The objective is to be the last player to place a tile on the board. |

So I altered my plan a bit and we explored Pick's Theorem. I laid out a lattice of dots on top of the shape and the student drew a polygon approximating the shape using the lattice points. We then counted the lattice points on the boundary and the points in the interior to calculate the area of the shape. We came up with an area of 42.5 square units (each square was 3 cm by 3 cm). |

## Planning and Building

## Scale

## Finance

## The Finished Project

I only built the front 1/3 of the house for reasons of both space and cost. When I have time, I'll build this model virtually in the LEGO Digital Designer. This tool allows you to design LEGO models in a virtual 3D environment and quickly see the number of bricks used to create it. My son is enjoying our LEGO house and currently has his LEGO fire truck rescuing our LEGO cat from the roof. New adventures await! |

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