Being on Twitter and following hashtags like #MTBoS and #ITeachMath allows me to see classroom mathematics well beyond my physical horizons. I get to glimpse creative and engaging mathematics education around the globe. Recently I saw a couple of different ideas that I've tried to adapt and apply for myself.
Since the Nova Scotia grade 8 classes are working on integer multiplication and division, I decided to create a math mystery of my own. Another nice source of math mysteries is the book Mathematical Team Games: Enjoyable Activities to Enhance the Curriculum by Vivien Lucas.
I liked this idea because it is relatively easy to create; just a find a series of questions with unique answers. Also, students get instant feedback. If their answer isn't on the map, they know they've made a mistake. I would call this purposeful practice as there is a goal to achieve at the end of the activity. There is a reason to persevere. Once students are familiar with the activity, you could give them a blank template (or they could hand draw their own version) and they could work in small groups to make their own treasure hunt activity (and answer key) and share it with each other.
The Role of Practice
I recently read Mark Chubb's (@MarkChubb3) blog post on the role of practice in math class. He discussed the differences between "rote practice" and "dynamic practice". Rote practice involves following procedures, drill and repetition while dynamic practice involves active student thinking, playful experiences and puzzles. I think that the Mystery activity is a more "dynamic" activity than doing the Treasure Hunt activity. However, I think that creating your own Treasure Hunt activity does involve additional characteristics of dynamic practice.
During a professional development session today with grade primary to grade 9 math coaches and support teachers, we spent some time working on solving some math puzzles. We used our work on these puzzles to reflect on what good group work looks and sounds like. We also discussed the characteristics of effective mathematical communication. It was great to see positive energy and teachers supporting and encouraging each other. Below are the five math puzzles and investigations that we worked on. We selected these puzzles because they are easy to explain, open to a wide range of students, and offered a fun challenge.
1-10 Card Investigation
This problem from Marilyn Burns asks you to find a special order for a stack of cards, Ace through 10. Place the stack of cards face down and turn over the first card. It should be the Ace. Place the next card on the bottom of the deck and then reveal the top card. It should be the number 2. Continue placing the next card on the bottom of the deck and then revealing the top card until all the cards are revealed. The face up cards should now be in order from Ace to 10. Marilyn recorded a video to demonstrate these rules that is much easier to understand than my written instructions. Marilyn's has a description of this problem on the MathSolutions website. This logic problem doesn't rely on any prerequisite mathematical knowledge and you can try it out quickly to see if you've come up with the correct solution. It gives students a chance to work together to try out strategies.
The Square-Sum Problem
Can you order all the numbers from 1 through 15 so that the sum of any two consecutive numbers are always a square number? For example, in the sequence: 4, 5, 11; 4+5=9 and 5+11=16.I really like this problem because there are some great extensions to take this problem farther and there is a very nice way to visualize the possible solutions. Numberphile has recorded a great video demonstrating both the problem and solution.
The Year Game - 2018
Use the digits in the year 2018 to write mathematical expressions for the counting numbers 1 through 100 (we only went to 20). Use any math operations (+, -, x, ÷, etc) and parentheses to write these expression. There is a more detailed description of this problem at the Math Forum website. For example, expressions for the number 1 might be: 10 ÷ (2+8) or 218^0. This problem is very similar to the classic Four Fours problem but with new digits each year. Which numbers are the hardest to find an expression for? Why do you think this is? I think this problem also leads nicely to a discussion about mathematical elegance and beauty. Look at a variety of expressions with the same value. Which expression do you think is the best? What makes for an "elegant" solution?
This problem challenged groups to make as many different shapes as possible with a perimeter of 12 units using a geoboard (or dot paper). Shapes were recorded on dot paper to make sure no shapes were repeated as reflections or rotations. I've seen variations of this problem in several places. One of them is Brian Bolt's book Mathematical Activities (1982). He suggests not only to find shapes with a perimeter of 12 but to also find the area of each shape. You can then find which shape has the maximum/minimum area. He also challenges students to find non-rectangular shapes (e.g. triangles) with a perimeter of 12 units. There were some good discussions about the lengths of diagonal line segments on the geoboard.
This problem is from NRICH. Finding a solution took some perseverance but most groups were eventually successful. The problem challenges you to create a square using 8 specific dominoes (0-1, 0-2, 0-3, 0-6, 1-2, 1-4, 2-2, and 3-5). In the 4 x 4 square, each of the columns and rows should sum to 8. The 3-5 domino seemed to be key as the rest of the squares in that row (or column) had to be blank.
What are your favourite math or logic puzzles?
This blog post could easily have been titled, "A Long Wait in a Really Long Line" but I like to be positive. I focus on the silver lining instead of the grey cloud. That's why I took a wait in a long long as an opportunity to practice estimation instead of a tedious and boring waste of time.
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.
Swimming in the hotel pool I saw these depth markers. As a math teacher, they made me a bit uneasy. What do you notice in the photos below? What do you wonder?
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.
Other Linear Conversions
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 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.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 10 M02 - Students will be expected to apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.
Mathematics at Work 10 M01 - Students will be expected to 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.
Mathematics Essentials 10 D1 - Demonstrate a working knowledge of the metric system and imperial system.
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.
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.
Sample Score Card
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?
Developing a Strategy
What makes a good score card? Here are some common student observations:
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.
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.
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 Rosette
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.
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.
Desmos Geometry Tool
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.
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 Lessons
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.
Super (Secret) Mathematics of Game Shows
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:
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.
My Lesson Planning Challenge
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.
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.
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.
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.
I was recently asked by a junior high school to support their math teachers to infuse opportunities for movement into their math classes. I've been brainstorming some ideas and I thought I would share them here. I'm including some general routines for including movement in meaningful ways as well as some examples of activities for specific math outcomes. I'm not including generic "brain breaks" or "movement breaks" which are short burst of physical activity designed to energize students.
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 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.
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.
Personal, professional and financial aspects of my life have come to a rare convergence which will allow me to attend the National Council of Teachers of Mathematics (NCTM) Annual conference this April in Washington D.C. This is the first time that I've had the opportunity to attend a conference like this. I've registered, arranged transportation and lodging, double checked my passport and I'm almost ready to go... just a few more steps.
Step 1 - Math Convention Carry
Packing my pencil case is something I put far too much thought into. This is what I currently have packed.
Step 2 - Pick Out Too Many Sessions
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.
Step 3 - Fine Tune My List of Sessions
Step 4 - Listen to Good Advice
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!