I spent a sunny Canada Day afternoon at the Halifax Commons. I went with my family to watch the SkyHawks, the Canadian Armed Forces Parachute Team. After the SkyHawks finished their jumps, we headed to the end of a a really long line so that my son could take a turn on a giant inflatable slide. Instead of dwelling on the length of the wait, we decided to focus on some fun estimation questions. *How many people do you think are in this line? *My son says he thinks it's more than 100 and I agree. *How long will it take to get to the front of the line?* I feel like we're in for a long wait. My initial estimate is at least 30 minutes. (Note: in order to answer this last question, I took a look at my watch to check the time... 12:06 pm).

Have we gotten any closer? It's 12:27 now (21 minutes in line) and it still feels like we've got a long way to go. There appears to be a strong correlation between the age of a child and the likelihood that they will have second thoughts at the top of the slide. This slows the line down dramatically as parents try to coax and cajole their child to make the leap. *How many people in this line are kids waiting to slide and how many are parents/guardians? What is the average age of the kids going down the slide?*

It's 12:40 now and we've been in line for over a half an hour. *How many steps do you estimate their are to the top of the slide? *I estimate that we're about half the distance to the slide from where we started. I realize that my initial estimate for how long we will be waiting was way off. At this point, I notice a group of upper elementary age students in front of us playing a hand clapping game called Concentration. After a bit, they shift to playing Chopsticks. This is a game I really like and I've used to introduce students to modular arithmetic so I watched their game to check out their strategy. It kept me entertained for a bit.

So close now I can taste it. It is 1:10 and we've been in line for over an hour. *How tall do you estimate that slide is?* I'd say it is about as tall as my house (a two story foursquare) which has a height of about 26 ft. (Note that the internet says this inflatable structure, called the 'Freestyle Combo', is 30 ft tall so my estimate seems pretty close).

We finally make it! It is 1:17 pm and we stood in line for 71 minutes. My son seems to think that this was a reasonable investment for such an awesome slide but I have my doubts. At least we got to do a lot of estimation while we waited in line. We definitely won't be heading to the back of the line for a second slide.

EL

]]>Just look at those significant digits. They look so precise. I first thought... going from the shallow end to the deep end, it gets 1 foot / 0.2 metres deeper. That must mean that 1 ft = 0.2 m right? But then if 1 ft is 0.2 m then shouldn't 3 ft in the shallow end be 0.6 m instead of 1.0 m?

So I looked at it another way... 1 m is the same as 3 ft... So 1 ft must be about 0.33 m. Which would make 4 ft equal to about 1.33 m not the 1.2 m as shown. But I know that a meter stick is shorter than a yard stick so this is just an approximation. No problem, they just rounded off both values.

Then I had a moment of doubt... in the shallow end the values are in a ratio of 1/3 and in the deep end the values are in a ratio of 4/12 which is also 1/3 so shouldn't this work out? Then I realized the errors and misconceptions in this line of thinking.

So looking at these signs I can find that 1 ft is either 0.2 m, 0.3 m or 0.33 repeating m. In case you're wondering, 1 metre is more accurately about 3.28084 feet. Given this, I really think the shallow end should say 0.9 metres instead of 1.0 metres (or perhaps the 3 ft should be 3 ft-3 inches instead). |

Today, as I was driving around, I looked more closely at the clearance signs that I passed under. There doesn't seem to be much consistency in the units used or precision. Do people with tall cars know the height of their car? I just know that I'm about 6 ft tall and my car is shorter than I am. Of course you can always just wing it. If you clear the warning bar, you're good to go. Anyway, I know for sure that my car is less than 11 foot 8 if I ever end up in North Carolina.

I also saw this at the airport. Is it exactly 6 cm or exactly 2.4 inches? They are not the same length although they are fairly close. They are about 1 mm different which is probably about the width of that thick orange line. I'm guessing that's not a coincidence. |

I saw this relatively accurate sign at a parking garage today so I took a photo. 6'0" is approximately 1.8288 metres so these values are the closest I've seen.

EL

]]>This 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.

Typically, 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?

What 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.

You 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.

EL

]]>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.

Nora 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

You 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.

After 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.

I'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

]]>Dan 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.

Sara 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.

One 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.

Shortly 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.

One 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

]]> 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). |

Next I showed students two different paper airplane designs; the Suzanne and the Classic Dart. I asked students to predict which would fly the farthest. I also asked how much difference, if any, they expected to see between the two designs. Most students predicted that the Suzanne would fly farthest. The next step was to create an experiment in order to test our predictions.

We split the class in half. Each half followed a specific set of instructions to fold one or the other of these planes (I had a handout with instructions for each design). We used different coloured paper for each design. Each student threw their plane three times and recorded each flight distance. We measured in feet since the floor tiles in the hallway were one square foot. The students then calculated their mean distance and shared this mean with their team. Each team then calculated a five number summary and sketched a box plot for their data.

Students declared the Suzanne to be the clear winner. The low ceiling height in the hallway seems to have favoured the glider design. We conjectured that the Dart may have performed better than Suzanne if they were thrown outdoors where students could throw at a higher launch angle. We also conjectured that the greater variation in the data for Suzanne was a result of the more complex folding required. Some planes were folded very well and others were a bit of a mess.

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. |

We finished class by watching a video of the world record throw for distance (we just watched the first 3 minutes of the video). The Suzanne, designed by John Collins and thrown by football quarterback Joe Ayoob holds the Guinness World Record for the farthest flight by a paper aircraft. The record throw was 226 feet, 10 inches (approx. 69.14 m). Our longest flight was just over 40 feet. Students seemed to really enjoy this activity. It allowed them to incorporate some movement in class and asked them to use mathematics and statistics in an authentic way to answer a real question.

**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.

EL

]]> 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. |

There are lots of different ways to incorporating movement into mathematics. Do you have additional suggestions of activities or resources for getting your junior high students moving around? Please consider sharing your ideas and I'll add them to this list.

EL

]]>Packing my pencil case is something I put far too much thought into. This is what I currently have packed.

- 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.

On my first pass through the list of sessions, I just pick out all the sessions that I'd like to attend. It turns out, there are a lot of sessions that I would really like to attend. Now to make some hard choices.

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. |

Speaking of advice, I was talking to a colleague yesterday about the NCTM conference. She said to not run from session to session for the whole conference. She said to slow down and talk to people. The networking and discussions between sessions can be very valuable. Returning with a few really good ideas can be better than returning with dozens that get lost in the shuffle.

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!

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!

EL

]]>My son and I were recently playing with a roll of tickets (a great math manipulative as it turns out). This led to some questions that appeared similar but were quite different in complexity. I took the pictures below and posted them to twitter.

The answer to the question on the right can be calculated with a single combination. You can use the "stars and bars" approach to think about the calculation required. First you have to decide if everyone gets a candy or not. If everyone gets at least one candy, then you can think of the problem as putting the twelve candies in a row (the "stars") and inserting 3 dividers in between them (the "bars") to divide the row into four sections. As an example, {★★★|★★|★★★|★★★★} would be one solution. Mathematically, there are 12 identical objects placed in 4 distinct bins, such that all bins contain at least one object. Given the 11 spaces between candies, how many ways are there to choose three of these spaces to place dividers.

For the same situation, if you allow each person to receive zero candies, there are more possibilities. Using the same "stars and bars" approach, you can think of all twelve candies and the three dividers and being placed in a row. How many ways are there to do this? There are a total of 12+3 spaces and either a candy or divider is placed into each one. As an example, {★★||★★★★★★|★★★★} would be one solution.

The answer to the question on the left regarding tearing tickets is actually a much more complicated question than the one on the right despite appearing very similar. In this question, we are separating identical objects into identical bins. This means that {★★|★★|★★|★★★★★★} is the same solution as {★★|★★★★★★|★★|★★} since they are both three groups of 2 and one group of 6. This type of problem involves partition numbers and they have been studied by mathematicians such as Leonhard Euler, Srinivasa Ramanujan and more recently Ken Ono. Partition numbers are an open area of mathematics research.

The solution for this problem is closely related to partition numbers. For every natural number *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 |

I recently tried out some problem sets from Craig Barton's SSDD problem website. SSDD stands for *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).

Michael Pershan wrote a blog post reflecting on the SSDD problem structure and how it might cause students to think in different ways. This type of reflection is why I write this blog and read other teachers blogs. Michael continued the conversation on Twitter and suggested that SSDD problems, “vary the deep differences while keeping the surface the same, and you draw attention to the way minor differences trigger different structure” I think that the type of Same and Different question prompt that I wrote about above also generates the same type of student thinking about the solution strategies required to solve a problem.

I think that the SSDD structure could lead to a variety of similar question routines. For example, you could give students variety of questions but instead of answering them, they could be asked to group the questions together that share a similar solution strategy. Or perhaps, you could give students a general context and ask them to create several different questions from this context connected to a variety of mathematics topics (similar to a Notice and Wonder strategy). I think it is exciting to have so much collaboration and thoughtful conversation online between mathematics educators.

EL

]]> 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. |

An engaging puzzle with simple instructions is to ask students to create a rectangle using pentomino pieces. Let them figure out what rectangle dimensions are possible given the total number of squares in the pentominoes they're using (for a full set of 12 pentominoes, 3x20, 4x15, 5x12 and 6x10 are possible). You can also print off a variety of other shapes and challenge students to fill those shapes with pentominoes or have students create their own shapes and challenge their friends to solve them. Younger students can be challenged to fill an empty picture frame or tray with pentominoes.

Here are five great games, puzzles and activities (in no particular order) that use pentominoes:

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. |

2. **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.

3. **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. |

4. **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. |

If you have a favourite activity, puzzle or game featuring pentominoes, I would love to hear about it!

EL

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